Research ArticleAn Entropy-Based Self-Adaptive Node Importance EvaluationMethod for Complex Networks

Qibo Sun Guoyu Yang and Ao Zhou

State Key Laboratory of Networking and Switching Technology Beijing University of Posts and TelecommunicationsBeijing China

Correspondence should be addressed to Qibo Sun qbsunbupteducn

Received 6 December 2019 Revised 25 February 2020 Accepted 10 March 2020 Published 25 April 2020

Guest Editor Yuan Yuan

Copyright copy 2020 Qibo Sun et al +is is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

Identifying important nodes in complex networks is essential in disease transmission control network attack protection andvaluable information detection Many evaluation indicators such as degree centrality betweenness centrality and closenesscentrality have been proposed to identify important nodes Some researchers assign different weight to different indicator andcombine them together to obtain the final evaluation results However the weight is usually subjectively assigned based on theresearcherrsquos experience which may lead to inaccurate results In this paper we propose an entropy-based self-adaptive nodeimportance evaluation method to evaluate node importance objectively Firstly based on complex network theory we select fourindicators to reflect different characteristics of the network structure Secondly we calculate the weights of different indicatorsbased on information entropy theory Finally based on aforesaid steps the node importance is obtained by weighted averagemethod +e experimental results show that our method performs better than the existing methods

1 Introduction

Complex network is playing an important role in our dailylife People communicate with each other and stay in touchwith old friends through online social networks +e In-ternet connects all the world so nowadays informationspreads faster and wider than before Electricity companiesbuild their own networks to provide electricity for pro-duction and living Policemen cooperate through their innernetworks in catching criminals However identifying im-portant nodes in complex networks is a critical issue invarious situations For example if the important nodes arewell isolated during disease transmission the epidemicwould be controlled effectively In traffic networks we canease traffic congestion by taking corresponding measures tosplit traffic flow in certain important nodes However howto evaluate node importance is not an easy task especially ifthere exist layered but complicated relationships instead of aflattened hierarchy A typical example is mobile edgecomputing network where smaller edge clouds connectswith each other but at the same time follow arrangements

from larger edge clouds While mobile users may connect toand accept service from both of them [1]

To solve this problem many researchers have proposeddifferent methods to identify important nodes in complexnetworks [2] Considering the local information of nodesand their neighbors degree centrality [3] semilocal cen-trality [4] and k-shell decomposition [5] are proposed tocharacterize the node importance But it does not take intoaccount the layer information among nodes of mobile edgecomputing network Because these indicators only considerthe local information of nodes the calculation complexity islow However they cannot accurately reflect the charac-teristics of the whole network To consider more globalcharacteristics of the whole network closeness centrality [6]betweenness centrality [7] etc are proposed +ese indi-cators consider paths and information flows +ereforethese indicators could reflect more characteristics about thewhole network structure However the calculation com-plexity is high To reduce the time complexity some re-searchers proposed methods that divide networks intoseveral parts such as community-based methods [8] and

HindawiComplexityVolume 2020 Article ID 4529429 13 pageshttpsdoiorg10115520204529429

cluster-based methods [9] Besides the aforesaid methodsresearchers also proposed methods considering not only thenumber of neighbors but also the importance of neighbors[10ndash13] +ese methods often iterate step by step to obtainthe steady result of each node

From the above discussions we can find that most of theindicators listed above could only reflect one characteristicof the network and cannot produce a comprehensiveevaluation To make up for it we can combine several in-dicators and assign different weights to different indicatorsto obtain better evaluation result However the weight as-signment is usually selected based on the subjective expe-rience of the researchers instead of enough scientific basiswhich has great possibility of leading to inaccurate evalu-ation results

In this paper we propose an information entropy the-ory-based self-adaptive node importance evaluation method(EBSAM) which could evaluate the node importance ob-jectively by adaptively assigning weights to different indi-cators Firstly we select four indicators to evaluate nodeimportance separately based on complex network theorySecondly the weights of different indicator are calculatedbased on the information entropy theory Finally the fourindicators are combined together to indicate the node im-portance So the proposed EBSAM can combine the best oneof other node importance indicators and compute theweights of them to create a more comprehensive indicatorTo study the effectiveness of our method we conductedexperiments on three different networks and compared theresults with other methods +e results show that the pro-posed method performs better than the existing methodsand improves the evaluation accuracy

+e rest of the paper is organized as follows Section 2gives a brief overview of the related work Section 3 illus-trates the problem definition and other basic preliminariesIn Section 4 we propose the entropy-based self-adaptiveevaluation method and explain the technical detail of it +ecomparative simulation experiments followed by the resultanalysis are given in Section 5 Section 6 concludes this paperand points out the future work

2 Related Work

Researchers have proposed many methods to evaluate thenode importance from different perspectives In this sectionwe look into some of the most recent and important researchworks done on node importance evaluation in complexnetworks

Liu et al [14] proposed a node ranking method based onthe importance of lines Firstly the proposed method cal-culates the importance of lines between nodes with theirtopological properties In addition the contribution of eachnode to the line importance is recorded +e final rankingresult is a combination of the node degree and its contri-bution to the linersquos importance Important bridge nodescould be well identified with lower computational com-plexity +e proposed method performs better than currentsingle local centrality measures but still does not considerenough global information for more accurate evaluation Hu

et al [15] applied the Locally Linear Embedding (LLE) al-gorithm [16] in evaluating node importance LLE which isoften used in machine learning is a nonlinear dimension-ality reduction technique In order to identify the importantnodes in a complex network several centrality measureshave been proposed +e input of the algorithm is a matrixconstructed by calculating the centrality measures of thenodes in the network However due to the limitation of LLEthis algorithm has some requirements for the distribution ofthe input data Xu et al [17] proposed a comprehensive nodeimportance evaluation approach by classifying nodes intoseveral types according to their functions in the networkDifferent measure indices are applied to evaluate the im-portance of different types of network nodes+e paper takesthe power transmission grids as example and divides nodesinto three types power supply node connection node andterminal load node For each type of node the ranking resultis obtained based on different centrality measures accordingto their function in the network Although this methodcould evaluate node importance precisely it is only appli-cable when nodes in the network could be divided intoseveral different types For networks where the functionalityof node is hard to distinguish the method performs badlyZhang et al [18] proposed a node importance evaluationmethod that combines betweenness centrality and closenesscentrality +ey believe that two types of factors determinethe importance of nodes +e first factor is its location in thenetwork and the second factor is the contribution of itsneighboring nodes Betweenness centrality has an importantimpact on the location of a node and closeness centralitycould determine the contribution of neighboring nodes +efinal node importance is a plus of the two factors Pinget al[19] believe that the importance contributions from both theadjacent and nonadjacent nodes have an important impacton the node importance +ey divide nodes into differentlayers according to their distance with the evaluated node Inaddition two parameters are defined to indicate the de-pendence strength between two nodes +e contributionprobability from one node to another is denoted by theimportance correlation parameter +e impact of the layeron the dependence strength is reflected by the strengthcorrelation parameter +e final result combines both theimportance of the evaluated node and the contribution ofother nodes in the network +e above methods mainlyexploit the local information or global information toevaluate the node importance

Yu et al [20] evaluate the node importance consideringboth the factors of the node closeness centrality degree andthe node degree +e global importance of nodes is repre-sented by closeness centrality+e local importance of nodesis characterized by the importance contribution betweenadjacent nodes +erefore both local attributes and globalattributes are considered during the node importanceevaluation process Hu et al [21] proposed a method thatcombines the k-shell decomposition algorithm with thecommunity centrality +e method considers not only thelocal information of the node but also the communitystructure it belongs to +e final result is a combination ofthese two indicators Different weights are assigned to the

2 Complexity

two indicators However the weight is set based on thepeoplersquos personal experience on network structure +ere-fore the evaluation result is very subjective Zhang et al [22]proposed a new algorithm combines betweenness centralityand Katz centrality +e proposed method comprehensivelyconsiders both the local node importance and the globalnode importance It overcomes the limitations of be-tweenness centrality for only considering shortest paths Inaddition it overcomes the limitations of Katz centrality forlocal optimum However the weights of the two indicatorsare selected by conducting amounts of experiments on thedataset with different weight values Apparently it is not agood way to determine the weight value by conducting lotsof experiments Yang and Xie [23] proposed a node im-portance evaluation method by using the multiobjectivedecision method +ey select several different representativeindicators +e weights of the indicators are calculated basedon Analytic Hierarchy Process Each node in the network isregarded as a solution and different indicators of each nodeare regarded as the solution properties +e evaluation resultis obtained through calculating the closeness degree of eachnode in the network to the ideal solution In this method theweights of different indicators are calculated using AnalyticHierarchy Process +erefore the accuracy is highly de-pendent on the researchersrsquo personal experience SimilarlyLiu et al [24] proposed a multiattribute ranking method fornode importance evaluation in complex networks +ey alsoselect four representative indicators and assign the weightsby using Analytic Hierarchy Process +e final result isobtained using the Technique for Order Preference bySimilarity to Ideal Object (TOPSIS)+emethod is similar tothe method proposed in [23] +e difference between thesetwo methods lies in representative indicators selection +eabove methods have the problem that the accuracy is highlydependent on the researchersrsquo personal experience

+erefore how to assign appropriate weights for dif-ferent indicators in different networks objectively andadaptively is still a problem to be solved We will address theproblem in this paper

3 System Model and NodeImportance Indicators

31 ampe Topology of Complex Networks +e complex net-works can be modelled as undirected and unweightednetworks We define an undirected and unweightednetwork as G V E V v1 v2 vn1113864 1113865 denotes theset of nodes in the complex network andE eij (vi vj) | i 1 n j 1 n1113966 1113967 denotes theset of edges in the complex network n is the totalnumber of nodes in the network

32 ampe Definition of Node Importance Indicators +ere aretwo different types of methods in network node importanceevaluation+e first type of methods only considers the localnode information which means that only the node itself andits neighborrsquos quantity are considered +e second type ofmethods considers the hierarchy infrastructure of a network

and the position of each node of the network which meansthat the global information of a node is considered Toabsorb their respective advantages and effectively evaluatethe node importance we adopt two local-information-re-lated attributions and two global-information-related at-tributions Degree centrality and improved K-shelldecomposition can reflect the local information of a nodeMoreover closeness centrality and betweenness centralitycan reflect the global information of a node

321 Degree Centrality Degree centrality [3] namely DC isdefined as the ratio of the number of edges that connect to anode directly

DCi di

n minus 1 (1)

where di is the number of edges connecting to node vi

directly n is the total number of nodes in the network Alarger value of DCi indicates that node vi has moreneighbors +erefore vi can influence more nodes in thenetwork and is more important

322 Closeness Centrality Closeness centrality (CC) [6] isdefined to represent the average distance of node vi to allother nodes in the network Suppose lij denotes the lengthof the shortest path from the source node vi to the desti-nation node vj +e average shortest distance from node vi

to all other nodes in the complex network can be calculatedby

si 1113936jneilij

n minus 1 (2)

+e smaller si is the more important vi is +e closenesscentrality CCi of node vi is defined as the reciprocal of si

CCi 1si

n minus 1

1113936jneilij (3)

If there is no path between vi and vj CCi is set to 0 Alarger value of CCi indicates that node vi is closer to thecentre of the network In other words the position of node vi

is very important in the network

323 Betweenness Centrality Betweenness centrality (BC)[7] is defined to represent the importance of a node in datatransmission Suppose vs and vt are two nodes in the net-work +e betweenness centrality is defined as follows

BCi 1113944ine sine tsne t

gist

gst

(4)

where gst denotes the number of the shortest paths from vs

to vt gist denotes the number of the shortest paths (from

node vs to node vt) that go through node vi A larger BCi

indicates that there more shortest paths travel through nodevi +erefore vi is more important in the data transmissionprocess

Complexity 3

324 Improved K-Shell Decomposition K-shell decompo-sition [5] is employed to identify the position of a node +eschematic diagram of K-shell decomposition is illustrated inFigure 1(a) Firstly remove all nodes whose degree is 1 fromthe network and set their Ks value to 1 Repeat this op-eration until the degree of all nodes in the network is largerthan 1 +en set Ks 2 3 and do the removing oper-ation continuously until all nodes have been removed fromthe network+e larger Ks is themore important the node isin the network

As can be seen from the definition K-shell decompo-sition would assign the same value to all nodes when thenetwork is a Star network or a Tree network To overcomethis challenge improved K-shell decomposition (IKs) isproposed by Liu et al in [24] +e process of improvedK-shell decomposition calculation is illustrated inFigure 1(b) Firstly IKs is initialized to 1+en all the nodeswhose degrees are minimum currently are removed from thenetwork and IKs is increased by 1 Repeat this operationuntil all nodes have been removed from the network +eimproved K-shell decomposition can overcome the limita-tion of K-shell decomposition and can reflect the charac-teristic of the network structure more precisely

4 Our Proposed Method

We illustrate the technical details of our entropy-based self-adaptive node importance evaluation method in this section

41 Attribute Matrix of Nodes +e nodes in a complexnetworks can be denoted by V v1 v2 vn1113864 1113865 +e indi-cators that are chosen to evaluate the node importance aredefined as I I1 I2 Ik1113864 1113865 k is the total number of in-dicators In our method I DCCCBC IKs and theattributes of node vi can be expressed as ai1 ai2 ai3 ai41113864 1113865+erefore the attribute matrix P is defined as follows

P

a11 a12 a13 a14

a21 a22 a23 a24

⋮ ⋮ ⋮ ⋮

an1 an2 an3 an4

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(5)

42 Data Normalization +e value of different indicatorscan vary in different ranges For example the value of DC is adecimal number in [0 1] while the value of IKs is larger than1 So the data should be normalized before they are com-bined together to allow for a uniform measurementCommon normalization methods include decimal scalingGaussian normalization zero-mean normalization min-max normalization etc Min-max normalization method isemployed to normalize the attribute matrix defined asfollowing

bij

aij minus mini1n

aij1113872 1113873

maxi1n

aij1113872 1113873 minus mini1n

aij1113872 1113873 (6)

+e normalized attribute matrix R is as follows

R

b11 b12 b13 b14

b21 b22 b23 b24

⋮ ⋮ ⋮ ⋮

bn1 bn2 bn3 bn4

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(7)

43 Weights Calculation Introduced by Claude E Shannonin 1948 entropy is a measure of unpredictability and un-certainty in information [25 26] For example the entropy iszero when we toss a two-headed coin+at is because there isa 100 chance of getting heads+e entropy has a maximumvalue when we toss a fair coin Since the chance of gettingtails is equal to the chance of getting heads there is no way topredict what will come next A smaller value of entropyindicates that there is less useful information content[27ndash31] In a multiattribute decision-making problem weneed to assign a larger weight to attribute with more usefulinformation rather than the attribute with greater uncer-tainty By analyzing the probability distribution of theoriginal data we could obtain the entropy objectivelyCalculating the weight of each attribute based on entropy ismore reasonable than setting it subjectively

In this paper node importance is decided by four in-dicators and their weights are obtained based on entropytheory Suppose the weight of each indicator is expressed asW w1 w2 w3 w41113864 1113865 According to Shannon entropy the-ory the entropy of each indicator can be calculated asfollows

pij bij

1113936ni1 bij

(8)

ej minus 1113936

ni1 pij lnpij

ln n (9)

where bij is the normalized jth indicator value of node viAnd ej(j 1 2 3 4) is the entropy of the indicators

As mentioned above the larger the entropy is the lessthe useful information contained in the indicator +ereforethe weight should be smaller +e weight of each indicator iscalculated by the following

wj 1 minus ej

4 minus 11139364j1 ej

(10)

We now illustrate the relationship between entropy andweight by taking the campus network of Beijing Universityof Posts and Telecommunications (BUPT) as an example+e topology of the campus network of Beijing University isillustrated in Figure 2 +e dots in Figure 2 denote the mainnodes of BUPT campus network +e relationship betweenentropy and weight is illustrated in Figure 3 As we can seethe larger the entropy is the smaller the weight is +esmaller the entropy is the more useful information can beprovided by an indicator +erefore the indicator withsmaller entropy has a larger weight

4 Complexity

44 Node Importance Ranking +e node importance iscalculated by the following

si 11139444

j1wjbij (11)

+e larger si(i 1 2 n) is the more important thenode is

+e general node importance calculation and noderanking steps in complex networks is shown in Algorithm 1Firstly determine the indicators DCCCBC IKs andcalculate the value of the four indicators for all nodes in thecomplex network +en we construct the attribute matrix P

based on equation (5) +irdly we calculate the normalizedattribute matrix R based on equations (6) and (7) Fourthlywe calculate the entropy e of each indicator based on

Ks = 1Ks = 2Ks = 3

(a)

IKs = 1IKs = 2IKs = 3

IKs = 4IKs = 5IKs = 6

(b)

Figure 1 (a) the schematic diagram of K-shell decomposition (b) the schematic diagram of improved K-shell decomposition

11

9

19

10

187

8

12

6

20

2624

29252327

28 30

22

36

35

34

21 31

32 33

37

14

13

16

15

17

2

1

3

4

5

Figure 2 Topological structure of BUPT campus network

Complexity 5

equations (8) and (9) (lines 1ndash11) +en the weights of allindicators could be obtained based on equation (10) (lines14ndash16) and the node importance is calculated based onequation (11) (lines 17ndash19) Finally all nodes are rankedbased on the node importance (lines 20ndash21) +e head of thenode list is the most importance node +e time complexityof our algorithm is O(n) where n denotes the number ofnodes in the network

5 Experiments

We conducted the experiments on three real networks andcompared the results of our method with the random se-lection method (Random) and the TOPSIS-RE method in

[24] +e experimental result proves that our method per-forms better

51 Experiment Setup +e selected networks are the campusnetwork of Beijing University of Posts and Telecommunications(BUPT) Shanxi Water Network and Shanxi Railway NetworkFirst we prove the effectiveness of ourmethod by experimentingon the BUPT campus network +en we illustrate the experi-mental results on Shanxi Water Network and Shanxi RailwayNetwork to see how the proposed method works in morecomplicated cases +e experiment is conducted on a PC withIntel Core i5-3470 32GHz CPU 4GB RAM

TOPSIS-RE extensively employs the Technique forOrder Preference by Similarity to Ideal Object (TOPSIS) to

00

02

04

06

08

10

12

DC CC BCCentrality measure

IKsEn

trop

y an

d w

eigh

t

EntropyWeight

Figure 3 +e relationship between entropy and weight

Input the normalized attribute matrix R

Output the ranking result(1) for each Ij in I do(2) sumj 0(3) for each vi in V do(4) sumj sumj + bij(5) end(6) ensumj 0(7) for each vi in V do(8) pij bijsumj(9) ensumj ensumj minus pij lnpij(10) end(11) ej ensumjln n

(12) esum esum + ej

(13) end(14) for each Ij in I do(15) wj 1 minus ej4 minus esum(16) end(17) for each vi in V do(18) si 1113936

4j1 wjbij

(19) end(20) Rank the node list based on si

(21) return the ranked node list

ALGORITHM 1 Node importance ranking algorithm

6 Complexity

evaluate the node importance +e core idea of TOPSIS-REis to construct a positive ideal object and a negative idealobject from the original data +e positive ideal object iscalculated based on the max value of the indicators and thenegative ideal object is calculated based on min value of the

indicators All methods are implemented by using thenetwork analysis software Cytoscape together with Javaprogramming language

In the experiments all nodes are ranked based on the nodeimportance+en the nodes are removed one by one from the

Table 1 Ranking result of BUPT campus network

Method Ranking resultsEBSAM 8 31 11 3 18 22 7 20 21 4 9 10 14 24 26 28TOPSIS-RE 8 18 31 3 11 7 21 20 9 10 4 14 24 26 28 22Random 19 12 23 25 30 32 13 34 16 5 27 21 9 20 6 10

6

18

9

19

20

10

12

7

282624

2725 2923

30

33

37

22

21

32

3435

36

13

16

15

14

17

1

4

2

5

(a)

6

109

19

12 282624

2725 2923

30

33

37

36

21

32

3435

13

16

15

14

17

1

4

2

5

(b)

6

109

19

121

5

2

282624

2725 2923

30

33

37

36

21

32

3435

13

16

15

14

17

(c)

65

2

19

121

2923

30

2725

33

37

36

21

32

3435

13

16

15

17

(d)

Figure 4 +e removing process with EBSAM on BUPTcampus network (a) 4 nodes have been removed (b) 8 nodes have been removed(c) 12 nodes have been removed (d) 16 nodes have been removed

02468

101214161820

1

NCC

3 5 7 9 11 13 15 17e number of nodes to be attacked

19 21 23 25 27 29 31 33 35 37

TOPSIS-REEBSAMRandom

Figure 5 Experiment results on BUPT campus network

Complexity 7

networks according to the ranking results +e Number ofConnected Components (NCC) is employed to evaluate theeffectiveness of the methods A connected component of anundirected network is a subgraph in which any two nodes areconnected to each other by edges After we remove one ormore nodes in a network the network will be divided intoseveral disconnected subgraphs Any node inside a subgraph isreachable from other nodes in the same subgraph +ere is nopath between two nodes belonging to different a subgraphNCC is the number of these disconnected subgraphs NCCreflects the connectivity of a network +e robustness of anetwork could be measured by calculating the size of thelargest connected component after removing a fraction of thenodes [32ndash34] +e number of connected component in anetwork could reflect its connectivity A larger value of NCCreflects that the network is divided into more disconnected

subgraphs which indicates the node you remove is moreimportant respect to network connectivity +erefore a largervalue of NCC indicates a better performance A node isconsidered to bemore important ifmore number of connectedcomponent increases after it has been removed

52 Experimental Results

521 Experiment Results on BUPT Campus Network+e topological structure of BUPT campus network is il-lustrated in Figure 2 +e number in each node is just anidentity of the node It does not have any meaning except toidentify different nodes +e node can be identified by itsnumber in the graph

+e node importance rank results of EBSAM TOPSIS-RE and a random selection algorithm (Random) are

Yun Cheng

Jin ChengLin Fen

Chang Zhi

Jin Zhong

Tai Yuan Yang Quan

Qi Zhou

Su Zhou

Da Tong

Figure 6 Shanxi water network

8 Complexity

0

5

10

15

20

25

30

35

1 3 5 7

NCC

9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43 45 47 49 51 53 55 57 59 61 63 65 67 69 71 73 75 77 79 81

TOPSIS-REEBSAMRandom

The number of nodes to be attacked

Figure 8 NCC of Shanxi water network

75

71

48

7473

81

68

8078

77

76

82

79

72

5655

70

69

57

54

51

5253

5049

44

33

4332

46

45

59

61 65

63

67

47

60

6664

62

58

16

35

27

14

28

34

221513

24 26

25

31

2021

10

12

2

9

1

38

36

29

39

1719

23

18

6

78

11

5

4

3

42

41

37

40

30

Figure 7 Topological structure of Shanxi water network

Complexity 9

illustrated in Table 1 We only list the top 16 nodes in therank results because the rank results of the rest nodes are thesame in EBSAM and TOPSIS-RE

According to the rank result we remove the nodes oneby one from the network until all nodes have been re-moved from the network We calculate the number ofconnected components in the network after removing anode +e removing process of EBSAM is shown in Fig-ure 4 We list out the topological structure of the networkafter removing every four nodes +e NCC of EBSAM andTOPSIS-RE and Random methods are shown in Figure 5As we can see the number of connected components ofRandom method is much less than the other two methods+erefore Random is less effective in destroying thenetwork by attacking the important nodes We can also seethat the number of connected components of EBSAM ismore than TOPSIS-RE Hence the connectivity of thenetwork is worse with EBSAM Attacking the network

based on the ranking result of EBSAM is more effectivethan TOPSIS-RE +at is because we obtain the weight ofthe four indicators objectively and adaptively other thanassign a fixed value subjectively

522 Experimental Results on Shanxi Water NetworkAs shown in Figure 6 Shanxi Water Network plays a vitalrole in the normal production and living activities+e greenline in Figure 6 denotes the water supply network +eShanxi water network provides guarantee for water demandof north China and its topological structure is shown inFigure 7 As shown in Figure 7 Shanxi Water Network iscomposed of 82 nodes +e experimental result is shown inFigure 8 As we can see the connectivity of the network hasbeen destroyed after the top 50 nodes have been attackedHowever NCC of our method is larger than the other twocompared methods +erefore the performance of EBSAMis better than other compared methods

Da Tong

Su Zhou

Qi Zhou

Tai Yuan Yang Quan

Jin Zhong

Lin Fen

Chang Zhi

Jin Cheng

Yun Cheng

The first 1000kmThe second 1000kmThe third 1000kmExpressway under construction and to be built

Figure 9 Shanxi railway network

10 Complexity

523 Experimental Results on Shanxi Railway NetworkFinally we conduct experiment on Shanxi Railway Net-work As shown in Figure 9 Shanxi Railway Network is apart of the transportation network in Shanxi It providesgreat convenience for peoplersquos outgoing and commodities

trading +e topological structure of Shanxi Water Net-work is shown in Figure 10 +e experimental result isshown in Figure 11 +e network is coming to break downafter the top 60 nodes have been attacked +e NCC ofShanxi Water Network obtains the largest ascent with our

4

31811

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3236

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886968

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Figure 10 Topological structure of Shanxi railway network

0

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1 5 9 13 17

NCC

21 25 29 33 37 41 45 49 53 57 61 65 69 73 77 81 85 89 93 97

TOPSIS-REEBSAMRandom

The number of nodes to be attacked

Figure 11 NCC of Shanxi railway network

Complexity 11

method +erefore EBSAM performs better than othermethods

6 Conclusions and Future Work

In this paper we proposed an entropy theory-based self-adaptive node importance evaluation method for complexnetworks Firstly we select four centrality measures whichcan reflect different characteristics of the node as nodeimportance evaluation indicators +en we combine themtogether with appropriate weights calculated by an entropytheory-based algorithm +e algorithm shows a strongadaptability and thus allows be widely implemented indifferent kinds of networks In the traditional method theweights are selected based on the subjective experience of theresearchers instead of enough scientific basis which wouldlead to inaccurate evaluation results +e proposed methodis better because it utilizes entropy theory to calculate theweight of each indicator A smaller value of entropy indicatesthat the corresponding attribution contains less useful in-formation In a multiattribute decision-making problem weneed to assign a larger weight to attribute with more usefulinformation rather than the attribute with greater uncer-tainty So with this algorithm we can better assign properweight to different attributions +e experimental results onthree types of real-world complex networks show that ourmethod performs better with compared methods Our on-going research will focus on investigating the effectiveness ofour method in more complex environments

Data Availability

+e data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

+e authors declare that they have no conflicts of interest

Acknowledgments

+is study was supported by National Key RampD Program ofChina (Funding No 2018YFB1402800) and the NaturalScience Foundation of China (No 61571066)

References

[1] SWang X Zhang Y Zhang LWang J Yang andWWangldquoA survey on mobile edge networks convergence of com-puting caching and communicationsrdquo Ieee Access vol 5pp 6757ndash6779 2017 (in English)

[2] X Ren and L Lu ldquoReview of ranking nodes in complexnetworksrdquo Chinese Science Bulletin vol 59 no 13pp 1175ndash1197 2014

[3] K L S Samanta and M Pal ldquoStudy on centrality measures insocial networks a surveyrdquo Social Network Analysis Miningvol 8 no 1 p 13 2018

[4] J Dong F Ye W Chen and J Wu ldquoIdentifying influentialnodes in complex networks via semi-local centralityrdquo inProceedings of the 2018 IEEE International Symposium on

Circuits and Systems (Iscas) (in English) Florence Italy May2018

[5] K Angelou M Maragakis and P Argyrakis ldquoA structuralanalysis of the patent citation network by the k-shell de-composition methodrdquo Physica A-Statistical Mechanics and ItsApplications vol 521 pp 476ndash483 2019 (in English)

[6] G Li M Li J Wang Y Li and Y Pan ldquoUnited neighborhoodcloseness centrality and orthology for predicting essentialproteinsrdquo IEEEACM Transactions on Computational Biologyand Bioinformatics 2018

[7] M Riondato and E Upfal ldquoABRA approximating be-tweenness centrality in static and dynamic graphs withrademacher averagesrdquo ACM Transactions on KnowledgeDiscovery from Data vol 12 no 5 pp 1ndash38 2018 (inEnglish)

[8] N Gupta A Singh and H Cherifi ldquoCentrality measures fornetworks with community structurerdquo Physica A StatisticalMechanics and Its Applications vol 452 pp 46ndash59 2016 (inEnglish)

[9] D Chen H Gao L Lu and T Zhou ldquoIdentifying influentialnodes in large-scale directed networks the role of clusteringrdquoPLoS One vol 8 Article ID e77455 no 10 (in English) 2013

[10] K Tu P Cui X Wang P Yu and W Zhu ldquoDeep recursivenetwork embedding with regular equivalencerdquo in Proceedingsof the Kddrsquo18 24th Acm Sigkdd International Conference onKnowledge Discovery amp Data Mining London UKpp 2357ndash2366 August 2018 (in English)

[11] R Poulin M-C Boily and B R Masse ldquoDynamical systemsto define centrality in social networksrdquo Social Networksvol 22 no 3 pp 187ndash220 2000 (in English)

[12] P Hu and T Mei ldquoRanking influential nodes in complexnetworks with structural holesrdquo Physica A Statistical Me-chanics and Its Applications vol 490 pp 624ndash631 15 2018 (inEnglish)

[13] L Lu Y Zhang C Yeung and T Zhou ldquoLeaders in socialnetworks the delicious caserdquo Plos One vol 6 no 6 (inEnglish) 2011

[14] J Liu Q Xiong W Shi X Shi and K Wang ldquoEvaluating theimportance of nodes in complex networksrdquo Physica A Sta-tistical Mechanics and Its Applications vol 452 pp 209ndash2192016 (in English)

[15] F Hu Y Liu and J Jin ldquoMulti-index evaluation algorithmbased on locally linear embedding for the node importance incomplex networksrdquo in Proceedings of ampirteenth InternationalSymposium on Distributed Computing and Applications toBusiness Engineering and Science (Dcabes 2014) pp 138ndash142(in English) Xian Ning China November 2014

[16] S T Roweis and L K Saul ldquoNonlinear dimensionality re-duction by locally linear embeddingrdquo Science vol 290no 5500 pp 2323ndash2326 2000 (in English)

[17] L Xu J Liu Y Liu Y Liu J Gou and B Masoud ldquoNodeimportance classified comprehensive assessmentrdquo Proceed-ings of the Csee vol 34 no 10 pp 1609ndash1617 2014

[18] X Zhang and X Wang ldquoEvaluation formula for communi-cation network node importancerdquo Journal of NortheasternUniversity vol 35 no 5 pp 663ndash666 2014

[19] H Ping W Fan and S Mei ldquoIdentifying node importance incomplex networksrdquo Physica A Statistical Mechanics and ItsApplications vol 429 pp 169ndash176 2015

[20] Y Yu ldquoNode importance measurement based on the degreeand closeness centralityrdquo Journal of Information and Com-putational Science vol 12 no 3 pp 1281ndash1291 2015

12 Complexity

[21] Q Hu Y Yin P Ma Y Gao Y Zhang and C Xing ldquoA newapproach to identify influential spreaders in complex net-worksrdquoActa Physica Sinica vol 62 no 14 (in Chinese) 2013

[22] Y Zhang Y Bao S Zhao J Chen and J Tang ldquoIdentifyingnode importance by combining betweenness centrality andKatz centralityrdquo in Proceedings of the 2015 InternationalConference on Cloud Computing and Big Data (Ccbd)pp 354ndash357 (in English) Shanghai China November 2015

[23] Y Yang and G Xie ldquoEfficient identification of node im-portance in social networksrdquo Information Processing ampManagement vol 52 no 5 pp 911ndash922 2016 (in English)

[24] Z Liu C Jiang J Wang and H Yu ldquo+e node importance inactual complex networks based on a multi-attribute rankingmethodrdquo Knowledge-Based Systems vol 84 pp 56ndash66 2015(in English)

[25] C E Shannon ldquoA mathematical theory of communicationrdquoBell System Technical Journal vol 27 no 4 pp 623ndash656 1948

[26] Q Zhang M Li and Y Deng ldquoMeasure the structure sim-ilarity of nodes in complex networks based on relative en-tropyrdquo Physica A Statistical Mechanics and Its Applicationsvol 491 pp 749ndash763 2018 (in English)

[27] Y He H Guo M Jin and P Ren ldquoA linguistic entropyweight method and its application in linguistic multi-attributegroup decision makingrdquo Nonlinear Dynamics vol 84 no 1pp 399ndash404 2016 (in English)

[28] X Liang and C Wei ldquoAn Atanassovrsquos intuitionistic fuzzymulti-attribute group decision making method based onentropy and similarity measurerdquo International Journal ofMachine Learning and Cybernetics vol 5 no 3 pp 435ndash4442014 (in English)

[29] T-Y Chen and C-H Li ldquoObjective weights with intuition-istic fuzzy entropy measures and computational experimentanalysisrdquo Applied Soft Computing vol 11 no 8 pp 5411ndash5423 2011 (in English)

[30] S-P Wan Q-Y Wang and J-Y Dong ldquo+e extendedVIKOR method for multi-attribute group decision makingwith triangular intuitionistic fuzzy numbersrdquo Knowledge-Based Systems vol 52 pp 65ndash77 2013 (in English)

[31] H Nguyen ldquoA new knowledge-based measure for intui-tionistic fuzzy sets and its application in multiple attributegroup decision makingrdquo Expert Systems with Applicationsvol 42 no 22 pp 8766ndash8774 2015 (in English)

[32] S Dereich and P Morters ldquoRandom networks with sublinearpreferential attachment the giant componentrdquoampe Annals ofProbability vol 41 no 1 pp 329ndash384 2013 (in English)

[33] C M Schneider A A Moreira J S Andrade S Havlin andH J Herrmann ldquoMitigation of malicious attacks on net-worksrdquo Proceedings of the National Academy of Sciences of theUnited States of America vol 108 no 10 pp 3838ndash3841 2011(in English)

[34] A Zareie and A Sheikhahmadi ldquoA hierarchical approach forinfluential node ranking in complex social networksrdquo ExpertSystems with Applications vol 93 pp 200ndash211 2018 (inEnglish)

Complexity 13

two indicators However the weight is set based on thepeoplersquos personal experience on network structure +ere-fore the evaluation result is very subjective Zhang et al [22]proposed a new algorithm combines betweenness centralityand Katz centrality +e proposed method comprehensivelyconsiders both the local node importance and the globalnode importance It overcomes the limitations of be-tweenness centrality for only considering shortest paths Inaddition it overcomes the limitations of Katz centrality forlocal optimum However the weights of the two indicatorsare selected by conducting amounts of experiments on thedataset with different weight values Apparently it is not agood way to determine the weight value by conducting lotsof experiments Yang and Xie [23] proposed a node im-portance evaluation method by using the multiobjectivedecision method +ey select several different representativeindicators +e weights of the indicators are calculated basedon Analytic Hierarchy Process Each node in the network isregarded as a solution and different indicators of each nodeare regarded as the solution properties +e evaluation resultis obtained through calculating the closeness degree of eachnode in the network to the ideal solution In this method theweights of different indicators are calculated using AnalyticHierarchy Process +erefore the accuracy is highly de-pendent on the researchersrsquo personal experience SimilarlyLiu et al [24] proposed a multiattribute ranking method fornode importance evaluation in complex networks +ey alsoselect four representative indicators and assign the weightsby using Analytic Hierarchy Process +e final result isobtained using the Technique for Order Preference bySimilarity to Ideal Object (TOPSIS)+emethod is similar tothe method proposed in [23] +e difference between thesetwo methods lies in representative indicators selection +eabove methods have the problem that the accuracy is highlydependent on the researchersrsquo personal experience

+erefore how to assign appropriate weights for dif-ferent indicators in different networks objectively andadaptively is still a problem to be solved We will address theproblem in this paper

3 System Model and NodeImportance Indicators

31 ampe Topology of Complex Networks +e complex net-works can be modelled as undirected and unweightednetworks We define an undirected and unweightednetwork as G V E V v1 v2 vn1113864 1113865 denotes theset of nodes in the complex network andE eij (vi vj) | i 1 n j 1 n1113966 1113967 denotes theset of edges in the complex network n is the totalnumber of nodes in the network

32 ampe Definition of Node Importance Indicators +ere aretwo different types of methods in network node importanceevaluation+e first type of methods only considers the localnode information which means that only the node itself andits neighborrsquos quantity are considered +e second type ofmethods considers the hierarchy infrastructure of a network

and the position of each node of the network which meansthat the global information of a node is considered Toabsorb their respective advantages and effectively evaluatethe node importance we adopt two local-information-re-lated attributions and two global-information-related at-tributions Degree centrality and improved K-shelldecomposition can reflect the local information of a nodeMoreover closeness centrality and betweenness centralitycan reflect the global information of a node

321 Degree Centrality Degree centrality [3] namely DC isdefined as the ratio of the number of edges that connect to anode directly

DCi di

n minus 1 (1)

where di is the number of edges connecting to node vi

directly n is the total number of nodes in the network Alarger value of DCi indicates that node vi has moreneighbors +erefore vi can influence more nodes in thenetwork and is more important

322 Closeness Centrality Closeness centrality (CC) [6] isdefined to represent the average distance of node vi to allother nodes in the network Suppose lij denotes the lengthof the shortest path from the source node vi to the desti-nation node vj +e average shortest distance from node vi

to all other nodes in the complex network can be calculatedby

si 1113936jneilij

n minus 1 (2)

+e smaller si is the more important vi is +e closenesscentrality CCi of node vi is defined as the reciprocal of si

CCi 1si

n minus 1

1113936jneilij (3)

If there is no path between vi and vj CCi is set to 0 Alarger value of CCi indicates that node vi is closer to thecentre of the network In other words the position of node vi

is very important in the network

323 Betweenness Centrality Betweenness centrality (BC)[7] is defined to represent the importance of a node in datatransmission Suppose vs and vt are two nodes in the net-work +e betweenness centrality is defined as follows

BCi 1113944ine sine tsne t

gist

gst

(4)

where gst denotes the number of the shortest paths from vs

to vt gist denotes the number of the shortest paths (from

node vs to node vt) that go through node vi A larger BCi

indicates that there more shortest paths travel through nodevi +erefore vi is more important in the data transmissionprocess

Complexity 3

324 Improved K-Shell Decomposition K-shell decompo-sition [5] is employed to identify the position of a node +eschematic diagram of K-shell decomposition is illustrated inFigure 1(a) Firstly remove all nodes whose degree is 1 fromthe network and set their Ks value to 1 Repeat this op-eration until the degree of all nodes in the network is largerthan 1 +en set Ks 2 3 and do the removing oper-ation continuously until all nodes have been removed fromthe network+e larger Ks is themore important the node isin the network

As can be seen from the definition K-shell decompo-sition would assign the same value to all nodes when thenetwork is a Star network or a Tree network To overcomethis challenge improved K-shell decomposition (IKs) isproposed by Liu et al in [24] +e process of improvedK-shell decomposition calculation is illustrated inFigure 1(b) Firstly IKs is initialized to 1+en all the nodeswhose degrees are minimum currently are removed from thenetwork and IKs is increased by 1 Repeat this operationuntil all nodes have been removed from the network +eimproved K-shell decomposition can overcome the limita-tion of K-shell decomposition and can reflect the charac-teristic of the network structure more precisely

4 Our Proposed Method

We illustrate the technical details of our entropy-based self-adaptive node importance evaluation method in this section

41 Attribute Matrix of Nodes +e nodes in a complexnetworks can be denoted by V v1 v2 vn1113864 1113865 +e indi-cators that are chosen to evaluate the node importance aredefined as I I1 I2 Ik1113864 1113865 k is the total number of in-dicators In our method I DCCCBC IKs and theattributes of node vi can be expressed as ai1 ai2 ai3 ai41113864 1113865+erefore the attribute matrix P is defined as follows

P

a11 a12 a13 a14

a21 a22 a23 a24

⋮ ⋮ ⋮ ⋮

an1 an2 an3 an4

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(5)

42 Data Normalization +e value of different indicatorscan vary in different ranges For example the value of DC is adecimal number in [0 1] while the value of IKs is larger than1 So the data should be normalized before they are com-bined together to allow for a uniform measurementCommon normalization methods include decimal scalingGaussian normalization zero-mean normalization min-max normalization etc Min-max normalization method isemployed to normalize the attribute matrix defined asfollowing

bij

aij minus mini1n

aij1113872 1113873

maxi1n

aij1113872 1113873 minus mini1n

aij1113872 1113873 (6)

+e normalized attribute matrix R is as follows

R

b11 b12 b13 b14

b21 b22 b23 b24

⋮ ⋮ ⋮ ⋮

bn1 bn2 bn3 bn4

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(7)

43 Weights Calculation Introduced by Claude E Shannonin 1948 entropy is a measure of unpredictability and un-certainty in information [25 26] For example the entropy iszero when we toss a two-headed coin+at is because there isa 100 chance of getting heads+e entropy has a maximumvalue when we toss a fair coin Since the chance of gettingtails is equal to the chance of getting heads there is no way topredict what will come next A smaller value of entropyindicates that there is less useful information content[27ndash31] In a multiattribute decision-making problem weneed to assign a larger weight to attribute with more usefulinformation rather than the attribute with greater uncer-tainty By analyzing the probability distribution of theoriginal data we could obtain the entropy objectivelyCalculating the weight of each attribute based on entropy ismore reasonable than setting it subjectively

In this paper node importance is decided by four in-dicators and their weights are obtained based on entropytheory Suppose the weight of each indicator is expressed asW w1 w2 w3 w41113864 1113865 According to Shannon entropy the-ory the entropy of each indicator can be calculated asfollows

pij bij

1113936ni1 bij

(8)

ej minus 1113936

ni1 pij lnpij

ln n (9)

where bij is the normalized jth indicator value of node viAnd ej(j 1 2 3 4) is the entropy of the indicators

As mentioned above the larger the entropy is the lessthe useful information contained in the indicator +ereforethe weight should be smaller +e weight of each indicator iscalculated by the following

wj 1 minus ej

4 minus 11139364j1 ej

(10)

We now illustrate the relationship between entropy andweight by taking the campus network of Beijing Universityof Posts and Telecommunications (BUPT) as an example+e topology of the campus network of Beijing University isillustrated in Figure 2 +e dots in Figure 2 denote the mainnodes of BUPT campus network +e relationship betweenentropy and weight is illustrated in Figure 3 As we can seethe larger the entropy is the smaller the weight is +esmaller the entropy is the more useful information can beprovided by an indicator +erefore the indicator withsmaller entropy has a larger weight

4 Complexity

44 Node Importance Ranking +e node importance iscalculated by the following

si 11139444

j1wjbij (11)

+e larger si(i 1 2 n) is the more important thenode is

+e general node importance calculation and noderanking steps in complex networks is shown in Algorithm 1Firstly determine the indicators DCCCBC IKs andcalculate the value of the four indicators for all nodes in thecomplex network +en we construct the attribute matrix P

based on equation (5) +irdly we calculate the normalizedattribute matrix R based on equations (6) and (7) Fourthlywe calculate the entropy e of each indicator based on

Ks = 1Ks = 2Ks = 3

(a)

IKs = 1IKs = 2IKs = 3

IKs = 4IKs = 5IKs = 6

(b)

Figure 1 (a) the schematic diagram of K-shell decomposition (b) the schematic diagram of improved K-shell decomposition

11

9

19

10

187

8

12

6

20

2624

29252327

28 30

22

36

35

34

21 31

32 33

37

14

13

16

15

17

2

1

3

4

5

Figure 2 Topological structure of BUPT campus network

Complexity 5

equations (8) and (9) (lines 1ndash11) +en the weights of allindicators could be obtained based on equation (10) (lines14ndash16) and the node importance is calculated based onequation (11) (lines 17ndash19) Finally all nodes are rankedbased on the node importance (lines 20ndash21) +e head of thenode list is the most importance node +e time complexityof our algorithm is O(n) where n denotes the number ofnodes in the network

5 Experiments

We conducted the experiments on three real networks andcompared the results of our method with the random se-lection method (Random) and the TOPSIS-RE method in

[24] +e experimental result proves that our method per-forms better

51 Experiment Setup +e selected networks are the campusnetwork of Beijing University of Posts and Telecommunications(BUPT) Shanxi Water Network and Shanxi Railway NetworkFirst we prove the effectiveness of ourmethod by experimentingon the BUPT campus network +en we illustrate the experi-mental results on Shanxi Water Network and Shanxi RailwayNetwork to see how the proposed method works in morecomplicated cases +e experiment is conducted on a PC withIntel Core i5-3470 32GHz CPU 4GB RAM

TOPSIS-RE extensively employs the Technique forOrder Preference by Similarity to Ideal Object (TOPSIS) to

00

02

04

06

08

10

12

DC CC BCCentrality measure

IKsEn

trop

y an

d w

eigh

t

EntropyWeight

Figure 3 +e relationship between entropy and weight

Input the normalized attribute matrix R

Output the ranking result(1) for each Ij in I do(2) sumj 0(3) for each vi in V do(4) sumj sumj + bij(5) end(6) ensumj 0(7) for each vi in V do(8) pij bijsumj(9) ensumj ensumj minus pij lnpij(10) end(11) ej ensumjln n

(12) esum esum + ej

(13) end(14) for each Ij in I do(15) wj 1 minus ej4 minus esum(16) end(17) for each vi in V do(18) si 1113936

4j1 wjbij

(19) end(20) Rank the node list based on si

(21) return the ranked node list

ALGORITHM 1 Node importance ranking algorithm

6 Complexity

evaluate the node importance +e core idea of TOPSIS-REis to construct a positive ideal object and a negative idealobject from the original data +e positive ideal object iscalculated based on the max value of the indicators and thenegative ideal object is calculated based on min value of the

indicators All methods are implemented by using thenetwork analysis software Cytoscape together with Javaprogramming language

In the experiments all nodes are ranked based on the nodeimportance+en the nodes are removed one by one from the

Table 1 Ranking result of BUPT campus network

Method Ranking resultsEBSAM 8 31 11 3 18 22 7 20 21 4 9 10 14 24 26 28TOPSIS-RE 8 18 31 3 11 7 21 20 9 10 4 14 24 26 28 22Random 19 12 23 25 30 32 13 34 16 5 27 21 9 20 6 10

6

18

9

19

20

10

12

7

282624

2725 2923

30

33

37

22

21

32

3435

36

13

16

15

14

17

1

4

2

5

(a)

6

109

19

12 282624

2725 2923

30

33

37

36

21

32

3435

13

16

15

14

17

1

4

2

5

(b)

6

109

19

121

5

2

282624

2725 2923

30

33

37

36

21

32

3435

13

16

15

14

17

(c)

65

2

19

121

2923

30

2725

33

37

36

21

32

3435

13

16

15

17

(d)

Figure 4 +e removing process with EBSAM on BUPTcampus network (a) 4 nodes have been removed (b) 8 nodes have been removed(c) 12 nodes have been removed (d) 16 nodes have been removed

02468

101214161820

1

NCC

3 5 7 9 11 13 15 17e number of nodes to be attacked

19 21 23 25 27 29 31 33 35 37

TOPSIS-REEBSAMRandom

Figure 5 Experiment results on BUPT campus network

Complexity 7

networks according to the ranking results +e Number ofConnected Components (NCC) is employed to evaluate theeffectiveness of the methods A connected component of anundirected network is a subgraph in which any two nodes areconnected to each other by edges After we remove one ormore nodes in a network the network will be divided intoseveral disconnected subgraphs Any node inside a subgraph isreachable from other nodes in the same subgraph +ere is nopath between two nodes belonging to different a subgraphNCC is the number of these disconnected subgraphs NCCreflects the connectivity of a network +e robustness of anetwork could be measured by calculating the size of thelargest connected component after removing a fraction of thenodes [32ndash34] +e number of connected component in anetwork could reflect its connectivity A larger value of NCCreflects that the network is divided into more disconnected

subgraphs which indicates the node you remove is moreimportant respect to network connectivity +erefore a largervalue of NCC indicates a better performance A node isconsidered to bemore important ifmore number of connectedcomponent increases after it has been removed

52 Experimental Results

521 Experiment Results on BUPT Campus Network+e topological structure of BUPT campus network is il-lustrated in Figure 2 +e number in each node is just anidentity of the node It does not have any meaning except toidentify different nodes +e node can be identified by itsnumber in the graph

+e node importance rank results of EBSAM TOPSIS-RE and a random selection algorithm (Random) are

Yun Cheng

Jin ChengLin Fen

Chang Zhi

Jin Zhong

Tai Yuan Yang Quan

Qi Zhou

Su Zhou

Da Tong

Figure 6 Shanxi water network

8 Complexity

0

5

10

15

20

25

30

35

1 3 5 7

NCC

9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43 45 47 49 51 53 55 57 59 61 63 65 67 69 71 73 75 77 79 81

TOPSIS-REEBSAMRandom

The number of nodes to be attacked

Figure 8 NCC of Shanxi water network

75

71

48

7473

81

68

8078

77

76

82

79

72

5655

70

69

57

54

51

5253

5049

44

33

4332

46

45

59

61 65

63

67

47

60

6664

62

58

16

35

27

14

28

34

221513

24 26

25

31

2021

10

12

2

9

1

38

36

29

39

1719

23

18

6

78

11

5

4

3

42

41

37

40

30

Figure 7 Topological structure of Shanxi water network

Complexity 9

illustrated in Table 1 We only list the top 16 nodes in therank results because the rank results of the rest nodes are thesame in EBSAM and TOPSIS-RE

According to the rank result we remove the nodes oneby one from the network until all nodes have been re-moved from the network We calculate the number ofconnected components in the network after removing anode +e removing process of EBSAM is shown in Fig-ure 4 We list out the topological structure of the networkafter removing every four nodes +e NCC of EBSAM andTOPSIS-RE and Random methods are shown in Figure 5As we can see the number of connected components ofRandom method is much less than the other two methods+erefore Random is less effective in destroying thenetwork by attacking the important nodes We can also seethat the number of connected components of EBSAM ismore than TOPSIS-RE Hence the connectivity of thenetwork is worse with EBSAM Attacking the network

based on the ranking result of EBSAM is more effectivethan TOPSIS-RE +at is because we obtain the weight ofthe four indicators objectively and adaptively other thanassign a fixed value subjectively

522 Experimental Results on Shanxi Water NetworkAs shown in Figure 6 Shanxi Water Network plays a vitalrole in the normal production and living activities+e greenline in Figure 6 denotes the water supply network +eShanxi water network provides guarantee for water demandof north China and its topological structure is shown inFigure 7 As shown in Figure 7 Shanxi Water Network iscomposed of 82 nodes +e experimental result is shown inFigure 8 As we can see the connectivity of the network hasbeen destroyed after the top 50 nodes have been attackedHowever NCC of our method is larger than the other twocompared methods +erefore the performance of EBSAMis better than other compared methods

Da Tong

Su Zhou

Qi Zhou

Tai Yuan Yang Quan

Jin Zhong

Lin Fen

Chang Zhi

Jin Cheng

Yun Cheng

The first 1000kmThe second 1000kmThe third 1000kmExpressway under construction and to be built

Figure 9 Shanxi railway network

10 Complexity

523 Experimental Results on Shanxi Railway NetworkFinally we conduct experiment on Shanxi Railway Net-work As shown in Figure 9 Shanxi Railway Network is apart of the transportation network in Shanxi It providesgreat convenience for peoplersquos outgoing and commodities

trading +e topological structure of Shanxi Water Net-work is shown in Figure 10 +e experimental result isshown in Figure 11 +e network is coming to break downafter the top 60 nodes have been attacked +e NCC ofShanxi Water Network obtains the largest ascent with our

4

31811

1

2

13

1219

2123

22

20

9

24

31

6

25

10

87

5

26

14

28

16 17

15

29

93

85

90

96

91

92

84

94

9597

98

33

27

30

3435

43

5150

3236

45

44

58

4138

40

3937

42

81

6465 66

7982

80 78

76

77

83

53

52

59

54

60

55

61

4948

46

47

86

62

87

63

57

89

56

886968

71

73 72

70

7475

67

Figure 10 Topological structure of Shanxi railway network

0

5

10

15

20

25

30

35

40

1 5 9 13 17

NCC

21 25 29 33 37 41 45 49 53 57 61 65 69 73 77 81 85 89 93 97

TOPSIS-REEBSAMRandom

The number of nodes to be attacked

Figure 11 NCC of Shanxi railway network

Complexity 11

method +erefore EBSAM performs better than othermethods

6 Conclusions and Future Work

In this paper we proposed an entropy theory-based self-adaptive node importance evaluation method for complexnetworks Firstly we select four centrality measures whichcan reflect different characteristics of the node as nodeimportance evaluation indicators +en we combine themtogether with appropriate weights calculated by an entropytheory-based algorithm +e algorithm shows a strongadaptability and thus allows be widely implemented indifferent kinds of networks In the traditional method theweights are selected based on the subjective experience of theresearchers instead of enough scientific basis which wouldlead to inaccurate evaluation results +e proposed methodis better because it utilizes entropy theory to calculate theweight of each indicator A smaller value of entropy indicatesthat the corresponding attribution contains less useful in-formation In a multiattribute decision-making problem weneed to assign a larger weight to attribute with more usefulinformation rather than the attribute with greater uncer-tainty So with this algorithm we can better assign properweight to different attributions +e experimental results onthree types of real-world complex networks show that ourmethod performs better with compared methods Our on-going research will focus on investigating the effectiveness ofour method in more complex environments

Data Availability

+e data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

+e authors declare that they have no conflicts of interest

Acknowledgments

+is study was supported by National Key RampD Program ofChina (Funding No 2018YFB1402800) and the NaturalScience Foundation of China (No 61571066)

References

[1] SWang X Zhang Y Zhang LWang J Yang andWWangldquoA survey on mobile edge networks convergence of com-puting caching and communicationsrdquo Ieee Access vol 5pp 6757ndash6779 2017 (in English)

[2] X Ren and L Lu ldquoReview of ranking nodes in complexnetworksrdquo Chinese Science Bulletin vol 59 no 13pp 1175ndash1197 2014

[3] K L S Samanta and M Pal ldquoStudy on centrality measures insocial networks a surveyrdquo Social Network Analysis Miningvol 8 no 1 p 13 2018

[4] J Dong F Ye W Chen and J Wu ldquoIdentifying influentialnodes in complex networks via semi-local centralityrdquo inProceedings of the 2018 IEEE International Symposium on

Circuits and Systems (Iscas) (in English) Florence Italy May2018

[5] K Angelou M Maragakis and P Argyrakis ldquoA structuralanalysis of the patent citation network by the k-shell de-composition methodrdquo Physica A-Statistical Mechanics and ItsApplications vol 521 pp 476ndash483 2019 (in English)

[6] G Li M Li J Wang Y Li and Y Pan ldquoUnited neighborhoodcloseness centrality and orthology for predicting essentialproteinsrdquo IEEEACM Transactions on Computational Biologyand Bioinformatics 2018

[7] M Riondato and E Upfal ldquoABRA approximating be-tweenness centrality in static and dynamic graphs withrademacher averagesrdquo ACM Transactions on KnowledgeDiscovery from Data vol 12 no 5 pp 1ndash38 2018 (inEnglish)

[8] N Gupta A Singh and H Cherifi ldquoCentrality measures fornetworks with community structurerdquo Physica A StatisticalMechanics and Its Applications vol 452 pp 46ndash59 2016 (inEnglish)

[9] D Chen H Gao L Lu and T Zhou ldquoIdentifying influentialnodes in large-scale directed networks the role of clusteringrdquoPLoS One vol 8 Article ID e77455 no 10 (in English) 2013

[10] K Tu P Cui X Wang P Yu and W Zhu ldquoDeep recursivenetwork embedding with regular equivalencerdquo in Proceedingsof the Kddrsquo18 24th Acm Sigkdd International Conference onKnowledge Discovery amp Data Mining London UKpp 2357ndash2366 August 2018 (in English)

[11] R Poulin M-C Boily and B R Masse ldquoDynamical systemsto define centrality in social networksrdquo Social Networksvol 22 no 3 pp 187ndash220 2000 (in English)

[12] P Hu and T Mei ldquoRanking influential nodes in complexnetworks with structural holesrdquo Physica A Statistical Me-chanics and Its Applications vol 490 pp 624ndash631 15 2018 (inEnglish)

[13] L Lu Y Zhang C Yeung and T Zhou ldquoLeaders in socialnetworks the delicious caserdquo Plos One vol 6 no 6 (inEnglish) 2011

[14] J Liu Q Xiong W Shi X Shi and K Wang ldquoEvaluating theimportance of nodes in complex networksrdquo Physica A Sta-tistical Mechanics and Its Applications vol 452 pp 209ndash2192016 (in English)

[15] F Hu Y Liu and J Jin ldquoMulti-index evaluation algorithmbased on locally linear embedding for the node importance incomplex networksrdquo in Proceedings of ampirteenth InternationalSymposium on Distributed Computing and Applications toBusiness Engineering and Science (Dcabes 2014) pp 138ndash142(in English) Xian Ning China November 2014

[16] S T Roweis and L K Saul ldquoNonlinear dimensionality re-duction by locally linear embeddingrdquo Science vol 290no 5500 pp 2323ndash2326 2000 (in English)

[17] L Xu J Liu Y Liu Y Liu J Gou and B Masoud ldquoNodeimportance classified comprehensive assessmentrdquo Proceed-ings of the Csee vol 34 no 10 pp 1609ndash1617 2014

[18] X Zhang and X Wang ldquoEvaluation formula for communi-cation network node importancerdquo Journal of NortheasternUniversity vol 35 no 5 pp 663ndash666 2014

[19] H Ping W Fan and S Mei ldquoIdentifying node importance incomplex networksrdquo Physica A Statistical Mechanics and ItsApplications vol 429 pp 169ndash176 2015

[20] Y Yu ldquoNode importance measurement based on the degreeand closeness centralityrdquo Journal of Information and Com-putational Science vol 12 no 3 pp 1281ndash1291 2015

12 Complexity

[21] Q Hu Y Yin P Ma Y Gao Y Zhang and C Xing ldquoA newapproach to identify influential spreaders in complex net-worksrdquoActa Physica Sinica vol 62 no 14 (in Chinese) 2013

[22] Y Zhang Y Bao S Zhao J Chen and J Tang ldquoIdentifyingnode importance by combining betweenness centrality andKatz centralityrdquo in Proceedings of the 2015 InternationalConference on Cloud Computing and Big Data (Ccbd)pp 354ndash357 (in English) Shanghai China November 2015

[23] Y Yang and G Xie ldquoEfficient identification of node im-portance in social networksrdquo Information Processing ampManagement vol 52 no 5 pp 911ndash922 2016 (in English)

[24] Z Liu C Jiang J Wang and H Yu ldquo+e node importance inactual complex networks based on a multi-attribute rankingmethodrdquo Knowledge-Based Systems vol 84 pp 56ndash66 2015(in English)

[25] C E Shannon ldquoA mathematical theory of communicationrdquoBell System Technical Journal vol 27 no 4 pp 623ndash656 1948

[26] Q Zhang M Li and Y Deng ldquoMeasure the structure sim-ilarity of nodes in complex networks based on relative en-tropyrdquo Physica A Statistical Mechanics and Its Applicationsvol 491 pp 749ndash763 2018 (in English)

[27] Y He H Guo M Jin and P Ren ldquoA linguistic entropyweight method and its application in linguistic multi-attributegroup decision makingrdquo Nonlinear Dynamics vol 84 no 1pp 399ndash404 2016 (in English)

[28] X Liang and C Wei ldquoAn Atanassovrsquos intuitionistic fuzzymulti-attribute group decision making method based onentropy and similarity measurerdquo International Journal ofMachine Learning and Cybernetics vol 5 no 3 pp 435ndash4442014 (in English)

[29] T-Y Chen and C-H Li ldquoObjective weights with intuition-istic fuzzy entropy measures and computational experimentanalysisrdquo Applied Soft Computing vol 11 no 8 pp 5411ndash5423 2011 (in English)

[30] S-P Wan Q-Y Wang and J-Y Dong ldquo+e extendedVIKOR method for multi-attribute group decision makingwith triangular intuitionistic fuzzy numbersrdquo Knowledge-Based Systems vol 52 pp 65ndash77 2013 (in English)

[31] H Nguyen ldquoA new knowledge-based measure for intui-tionistic fuzzy sets and its application in multiple attributegroup decision makingrdquo Expert Systems with Applicationsvol 42 no 22 pp 8766ndash8774 2015 (in English)

[32] S Dereich and P Morters ldquoRandom networks with sublinearpreferential attachment the giant componentrdquoampe Annals ofProbability vol 41 no 1 pp 329ndash384 2013 (in English)

[33] C M Schneider A A Moreira J S Andrade S Havlin andH J Herrmann ldquoMitigation of malicious attacks on net-worksrdquo Proceedings of the National Academy of Sciences of theUnited States of America vol 108 no 10 pp 3838ndash3841 2011(in English)

[34] A Zareie and A Sheikhahmadi ldquoA hierarchical approach forinfluential node ranking in complex social networksrdquo ExpertSystems with Applications vol 93 pp 200ndash211 2018 (inEnglish)

Complexity 13

324 Improved K-Shell Decomposition K-shell decompo-sition [5] is employed to identify the position of a node +eschematic diagram of K-shell decomposition is illustrated inFigure 1(a) Firstly remove all nodes whose degree is 1 fromthe network and set their Ks value to 1 Repeat this op-eration until the degree of all nodes in the network is largerthan 1 +en set Ks 2 3 and do the removing oper-ation continuously until all nodes have been removed fromthe network+e larger Ks is themore important the node isin the network

As can be seen from the definition K-shell decompo-sition would assign the same value to all nodes when thenetwork is a Star network or a Tree network To overcomethis challenge improved K-shell decomposition (IKs) isproposed by Liu et al in [24] +e process of improvedK-shell decomposition calculation is illustrated inFigure 1(b) Firstly IKs is initialized to 1+en all the nodeswhose degrees are minimum currently are removed from thenetwork and IKs is increased by 1 Repeat this operationuntil all nodes have been removed from the network +eimproved K-shell decomposition can overcome the limita-tion of K-shell decomposition and can reflect the charac-teristic of the network structure more precisely

4 Our Proposed Method

We illustrate the technical details of our entropy-based self-adaptive node importance evaluation method in this section

41 Attribute Matrix of Nodes +e nodes in a complexnetworks can be denoted by V v1 v2 vn1113864 1113865 +e indi-cators that are chosen to evaluate the node importance aredefined as I I1 I2 Ik1113864 1113865 k is the total number of in-dicators In our method I DCCCBC IKs and theattributes of node vi can be expressed as ai1 ai2 ai3 ai41113864 1113865+erefore the attribute matrix P is defined as follows

P

a11 a12 a13 a14

a21 a22 a23 a24

⋮ ⋮ ⋮ ⋮

an1 an2 an3 an4

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(5)

42 Data Normalization +e value of different indicatorscan vary in different ranges For example the value of DC is adecimal number in [0 1] while the value of IKs is larger than1 So the data should be normalized before they are com-bined together to allow for a uniform measurementCommon normalization methods include decimal scalingGaussian normalization zero-mean normalization min-max normalization etc Min-max normalization method isemployed to normalize the attribute matrix defined asfollowing

bij

aij minus mini1n

aij1113872 1113873

maxi1n

aij1113872 1113873 minus mini1n

aij1113872 1113873 (6)

+e normalized attribute matrix R is as follows

R

b11 b12 b13 b14

b21 b22 b23 b24

⋮ ⋮ ⋮ ⋮

bn1 bn2 bn3 bn4

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(7)

43 Weights Calculation Introduced by Claude E Shannonin 1948 entropy is a measure of unpredictability and un-certainty in information [25 26] For example the entropy iszero when we toss a two-headed coin+at is because there isa 100 chance of getting heads+e entropy has a maximumvalue when we toss a fair coin Since the chance of gettingtails is equal to the chance of getting heads there is no way topredict what will come next A smaller value of entropyindicates that there is less useful information content[27ndash31] In a multiattribute decision-making problem weneed to assign a larger weight to attribute with more usefulinformation rather than the attribute with greater uncer-tainty By analyzing the probability distribution of theoriginal data we could obtain the entropy objectivelyCalculating the weight of each attribute based on entropy ismore reasonable than setting it subjectively

In this paper node importance is decided by four in-dicators and their weights are obtained based on entropytheory Suppose the weight of each indicator is expressed asW w1 w2 w3 w41113864 1113865 According to Shannon entropy the-ory the entropy of each indicator can be calculated asfollows

pij bij

1113936ni1 bij

(8)

ej minus 1113936

ni1 pij lnpij

ln n (9)

where bij is the normalized jth indicator value of node viAnd ej(j 1 2 3 4) is the entropy of the indicators

As mentioned above the larger the entropy is the lessthe useful information contained in the indicator +ereforethe weight should be smaller +e weight of each indicator iscalculated by the following

wj 1 minus ej

4 minus 11139364j1 ej

(10)

We now illustrate the relationship between entropy andweight by taking the campus network of Beijing Universityof Posts and Telecommunications (BUPT) as an example+e topology of the campus network of Beijing University isillustrated in Figure 2 +e dots in Figure 2 denote the mainnodes of BUPT campus network +e relationship betweenentropy and weight is illustrated in Figure 3 As we can seethe larger the entropy is the smaller the weight is +esmaller the entropy is the more useful information can beprovided by an indicator +erefore the indicator withsmaller entropy has a larger weight

4 Complexity

44 Node Importance Ranking +e node importance iscalculated by the following

si 11139444

j1wjbij (11)

+e larger si(i 1 2 n) is the more important thenode is

+e general node importance calculation and noderanking steps in complex networks is shown in Algorithm 1Firstly determine the indicators DCCCBC IKs andcalculate the value of the four indicators for all nodes in thecomplex network +en we construct the attribute matrix P

based on equation (5) +irdly we calculate the normalizedattribute matrix R based on equations (6) and (7) Fourthlywe calculate the entropy e of each indicator based on

Ks = 1Ks = 2Ks = 3

(a)

IKs = 1IKs = 2IKs = 3

IKs = 4IKs = 5IKs = 6

(b)

Figure 1 (a) the schematic diagram of K-shell decomposition (b) the schematic diagram of improved K-shell decomposition

11

9

19

10

187

8

12

6

20

2624

29252327

28 30

22

36

35

34

21 31

32 33

37

14

13

16

15

17

2

1

3

4

5

Figure 2 Topological structure of BUPT campus network

Complexity 5

equations (8) and (9) (lines 1ndash11) +en the weights of allindicators could be obtained based on equation (10) (lines14ndash16) and the node importance is calculated based onequation (11) (lines 17ndash19) Finally all nodes are rankedbased on the node importance (lines 20ndash21) +e head of thenode list is the most importance node +e time complexityof our algorithm is O(n) where n denotes the number ofnodes in the network

5 Experiments

We conducted the experiments on three real networks andcompared the results of our method with the random se-lection method (Random) and the TOPSIS-RE method in

[24] +e experimental result proves that our method per-forms better

51 Experiment Setup +e selected networks are the campusnetwork of Beijing University of Posts and Telecommunications(BUPT) Shanxi Water Network and Shanxi Railway NetworkFirst we prove the effectiveness of ourmethod by experimentingon the BUPT campus network +en we illustrate the experi-mental results on Shanxi Water Network and Shanxi RailwayNetwork to see how the proposed method works in morecomplicated cases +e experiment is conducted on a PC withIntel Core i5-3470 32GHz CPU 4GB RAM

TOPSIS-RE extensively employs the Technique forOrder Preference by Similarity to Ideal Object (TOPSIS) to

00

02

04

06

08

10

12

DC CC BCCentrality measure

IKsEn

trop

y an

d w

eigh

t

EntropyWeight

Figure 3 +e relationship between entropy and weight

Input the normalized attribute matrix R

Output the ranking result(1) for each Ij in I do(2) sumj 0(3) for each vi in V do(4) sumj sumj + bij(5) end(6) ensumj 0(7) for each vi in V do(8) pij bijsumj(9) ensumj ensumj minus pij lnpij(10) end(11) ej ensumjln n

(12) esum esum + ej

(13) end(14) for each Ij in I do(15) wj 1 minus ej4 minus esum(16) end(17) for each vi in V do(18) si 1113936

4j1 wjbij

(19) end(20) Rank the node list based on si

(21) return the ranked node list

ALGORITHM 1 Node importance ranking algorithm

6 Complexity

evaluate the node importance +e core idea of TOPSIS-REis to construct a positive ideal object and a negative idealobject from the original data +e positive ideal object iscalculated based on the max value of the indicators and thenegative ideal object is calculated based on min value of the

indicators All methods are implemented by using thenetwork analysis software Cytoscape together with Javaprogramming language

In the experiments all nodes are ranked based on the nodeimportance+en the nodes are removed one by one from the

Table 1 Ranking result of BUPT campus network

Method Ranking resultsEBSAM 8 31 11 3 18 22 7 20 21 4 9 10 14 24 26 28TOPSIS-RE 8 18 31 3 11 7 21 20 9 10 4 14 24 26 28 22Random 19 12 23 25 30 32 13 34 16 5 27 21 9 20 6 10

6

18

9

19

20

10

12

7

282624

2725 2923

30

33

37

22

21

32

3435

36

13

16

15

14

17

1

4

2

5

(a)

6

109

19

12 282624

2725 2923

30

33

37

36

21

32

3435

13

16

15

14

17

1

4

2

5

(b)

6

109

19

121

5

2

282624

2725 2923

30

33

37

36

21

32

3435

13

16

15

14

17

(c)

65

2

19

121

2923

30

2725

33

37

36

21

32

3435

13

16

15

17

(d)

Figure 4 +e removing process with EBSAM on BUPTcampus network (a) 4 nodes have been removed (b) 8 nodes have been removed(c) 12 nodes have been removed (d) 16 nodes have been removed

02468

101214161820

1

NCC

3 5 7 9 11 13 15 17e number of nodes to be attacked

19 21 23 25 27 29 31 33 35 37

TOPSIS-REEBSAMRandom

Figure 5 Experiment results on BUPT campus network

Complexity 7

networks according to the ranking results +e Number ofConnected Components (NCC) is employed to evaluate theeffectiveness of the methods A connected component of anundirected network is a subgraph in which any two nodes areconnected to each other by edges After we remove one ormore nodes in a network the network will be divided intoseveral disconnected subgraphs Any node inside a subgraph isreachable from other nodes in the same subgraph +ere is nopath between two nodes belonging to different a subgraphNCC is the number of these disconnected subgraphs NCCreflects the connectivity of a network +e robustness of anetwork could be measured by calculating the size of thelargest connected component after removing a fraction of thenodes [32ndash34] +e number of connected component in anetwork could reflect its connectivity A larger value of NCCreflects that the network is divided into more disconnected

subgraphs which indicates the node you remove is moreimportant respect to network connectivity +erefore a largervalue of NCC indicates a better performance A node isconsidered to bemore important ifmore number of connectedcomponent increases after it has been removed

52 Experimental Results

521 Experiment Results on BUPT Campus Network+e topological structure of BUPT campus network is il-lustrated in Figure 2 +e number in each node is just anidentity of the node It does not have any meaning except toidentify different nodes +e node can be identified by itsnumber in the graph

+e node importance rank results of EBSAM TOPSIS-RE and a random selection algorithm (Random) are

Yun Cheng

Jin ChengLin Fen

Chang Zhi

Jin Zhong

Tai Yuan Yang Quan

Qi Zhou

Su Zhou

Da Tong

Figure 6 Shanxi water network

8 Complexity

0

5

10

15

20

25

30

35

1 3 5 7

NCC

9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43 45 47 49 51 53 55 57 59 61 63 65 67 69 71 73 75 77 79 81

TOPSIS-REEBSAMRandom

The number of nodes to be attacked

Figure 8 NCC of Shanxi water network

75

71

48

7473

81

68

8078

77

76

82

79

72

5655

70

69

57

54

51

5253

5049

44

33

4332

46

45

59

61 65

63

67

47

60

6664

62

58

16

35

27

14

28

34

221513

24 26

25

31

2021

10

12

2

9

1

38

36

29

39

1719

23

18

6

78

11

5

4

3

42

41

37

40

30

Figure 7 Topological structure of Shanxi water network

Complexity 9

illustrated in Table 1 We only list the top 16 nodes in therank results because the rank results of the rest nodes are thesame in EBSAM and TOPSIS-RE

According to the rank result we remove the nodes oneby one from the network until all nodes have been re-moved from the network We calculate the number ofconnected components in the network after removing anode +e removing process of EBSAM is shown in Fig-ure 4 We list out the topological structure of the networkafter removing every four nodes +e NCC of EBSAM andTOPSIS-RE and Random methods are shown in Figure 5As we can see the number of connected components ofRandom method is much less than the other two methods+erefore Random is less effective in destroying thenetwork by attacking the important nodes We can also seethat the number of connected components of EBSAM ismore than TOPSIS-RE Hence the connectivity of thenetwork is worse with EBSAM Attacking the network

based on the ranking result of EBSAM is more effectivethan TOPSIS-RE +at is because we obtain the weight ofthe four indicators objectively and adaptively other thanassign a fixed value subjectively

522 Experimental Results on Shanxi Water NetworkAs shown in Figure 6 Shanxi Water Network plays a vitalrole in the normal production and living activities+e greenline in Figure 6 denotes the water supply network +eShanxi water network provides guarantee for water demandof north China and its topological structure is shown inFigure 7 As shown in Figure 7 Shanxi Water Network iscomposed of 82 nodes +e experimental result is shown inFigure 8 As we can see the connectivity of the network hasbeen destroyed after the top 50 nodes have been attackedHowever NCC of our method is larger than the other twocompared methods +erefore the performance of EBSAMis better than other compared methods

Da Tong

Su Zhou

Qi Zhou

Tai Yuan Yang Quan

Jin Zhong

Lin Fen

Chang Zhi

Jin Cheng

Yun Cheng

The first 1000kmThe second 1000kmThe third 1000kmExpressway under construction and to be built

Figure 9 Shanxi railway network

10 Complexity

523 Experimental Results on Shanxi Railway NetworkFinally we conduct experiment on Shanxi Railway Net-work As shown in Figure 9 Shanxi Railway Network is apart of the transportation network in Shanxi It providesgreat convenience for peoplersquos outgoing and commodities

trading +e topological structure of Shanxi Water Net-work is shown in Figure 10 +e experimental result isshown in Figure 11 +e network is coming to break downafter the top 60 nodes have been attacked +e NCC ofShanxi Water Network obtains the largest ascent with our

4

31811

1

2

13

1219

2123

22

20

9

24

31

6

25

10

87

5

26

14

28

16 17

15

29

93

85

90

96

91

92

84

94

9597

98

33

27

30

3435

43

5150

3236

45

44

58

4138

40

3937

42

81

6465 66

7982

80 78

76

77

83

53

52

59

54

60

55

61

4948

46

47

86

62

87

63

57

89

56

886968

71

73 72

70

7475

67

Figure 10 Topological structure of Shanxi railway network

0

5

10

15

20

25

30

35

40

1 5 9 13 17

NCC

21 25 29 33 37 41 45 49 53 57 61 65 69 73 77 81 85 89 93 97

TOPSIS-REEBSAMRandom

The number of nodes to be attacked

Figure 11 NCC of Shanxi railway network

Complexity 11

method +erefore EBSAM performs better than othermethods

6 Conclusions and Future Work

In this paper we proposed an entropy theory-based self-adaptive node importance evaluation method for complexnetworks Firstly we select four centrality measures whichcan reflect different characteristics of the node as nodeimportance evaluation indicators +en we combine themtogether with appropriate weights calculated by an entropytheory-based algorithm +e algorithm shows a strongadaptability and thus allows be widely implemented indifferent kinds of networks In the traditional method theweights are selected based on the subjective experience of theresearchers instead of enough scientific basis which wouldlead to inaccurate evaluation results +e proposed methodis better because it utilizes entropy theory to calculate theweight of each indicator A smaller value of entropy indicatesthat the corresponding attribution contains less useful in-formation In a multiattribute decision-making problem weneed to assign a larger weight to attribute with more usefulinformation rather than the attribute with greater uncer-tainty So with this algorithm we can better assign properweight to different attributions +e experimental results onthree types of real-world complex networks show that ourmethod performs better with compared methods Our on-going research will focus on investigating the effectiveness ofour method in more complex environments

Data Availability

+e data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

+e authors declare that they have no conflicts of interest

Acknowledgments

+is study was supported by National Key RampD Program ofChina (Funding No 2018YFB1402800) and the NaturalScience Foundation of China (No 61571066)

References

[1] SWang X Zhang Y Zhang LWang J Yang andWWangldquoA survey on mobile edge networks convergence of com-puting caching and communicationsrdquo Ieee Access vol 5pp 6757ndash6779 2017 (in English)

[2] X Ren and L Lu ldquoReview of ranking nodes in complexnetworksrdquo Chinese Science Bulletin vol 59 no 13pp 1175ndash1197 2014

[3] K L S Samanta and M Pal ldquoStudy on centrality measures insocial networks a surveyrdquo Social Network Analysis Miningvol 8 no 1 p 13 2018

[4] J Dong F Ye W Chen and J Wu ldquoIdentifying influentialnodes in complex networks via semi-local centralityrdquo inProceedings of the 2018 IEEE International Symposium on

Circuits and Systems (Iscas) (in English) Florence Italy May2018

[5] K Angelou M Maragakis and P Argyrakis ldquoA structuralanalysis of the patent citation network by the k-shell de-composition methodrdquo Physica A-Statistical Mechanics and ItsApplications vol 521 pp 476ndash483 2019 (in English)

[6] G Li M Li J Wang Y Li and Y Pan ldquoUnited neighborhoodcloseness centrality and orthology for predicting essentialproteinsrdquo IEEEACM Transactions on Computational Biologyand Bioinformatics 2018

[7] M Riondato and E Upfal ldquoABRA approximating be-tweenness centrality in static and dynamic graphs withrademacher averagesrdquo ACM Transactions on KnowledgeDiscovery from Data vol 12 no 5 pp 1ndash38 2018 (inEnglish)

[8] N Gupta A Singh and H Cherifi ldquoCentrality measures fornetworks with community structurerdquo Physica A StatisticalMechanics and Its Applications vol 452 pp 46ndash59 2016 (inEnglish)

[9] D Chen H Gao L Lu and T Zhou ldquoIdentifying influentialnodes in large-scale directed networks the role of clusteringrdquoPLoS One vol 8 Article ID e77455 no 10 (in English) 2013

[10] K Tu P Cui X Wang P Yu and W Zhu ldquoDeep recursivenetwork embedding with regular equivalencerdquo in Proceedingsof the Kddrsquo18 24th Acm Sigkdd International Conference onKnowledge Discovery amp Data Mining London UKpp 2357ndash2366 August 2018 (in English)

[11] R Poulin M-C Boily and B R Masse ldquoDynamical systemsto define centrality in social networksrdquo Social Networksvol 22 no 3 pp 187ndash220 2000 (in English)

[12] P Hu and T Mei ldquoRanking influential nodes in complexnetworks with structural holesrdquo Physica A Statistical Me-chanics and Its Applications vol 490 pp 624ndash631 15 2018 (inEnglish)

[13] L Lu Y Zhang C Yeung and T Zhou ldquoLeaders in socialnetworks the delicious caserdquo Plos One vol 6 no 6 (inEnglish) 2011

[14] J Liu Q Xiong W Shi X Shi and K Wang ldquoEvaluating theimportance of nodes in complex networksrdquo Physica A Sta-tistical Mechanics and Its Applications vol 452 pp 209ndash2192016 (in English)

[15] F Hu Y Liu and J Jin ldquoMulti-index evaluation algorithmbased on locally linear embedding for the node importance incomplex networksrdquo in Proceedings of ampirteenth InternationalSymposium on Distributed Computing and Applications toBusiness Engineering and Science (Dcabes 2014) pp 138ndash142(in English) Xian Ning China November 2014

[16] S T Roweis and L K Saul ldquoNonlinear dimensionality re-duction by locally linear embeddingrdquo Science vol 290no 5500 pp 2323ndash2326 2000 (in English)

[17] L Xu J Liu Y Liu Y Liu J Gou and B Masoud ldquoNodeimportance classified comprehensive assessmentrdquo Proceed-ings of the Csee vol 34 no 10 pp 1609ndash1617 2014

[18] X Zhang and X Wang ldquoEvaluation formula for communi-cation network node importancerdquo Journal of NortheasternUniversity vol 35 no 5 pp 663ndash666 2014

[19] H Ping W Fan and S Mei ldquoIdentifying node importance incomplex networksrdquo Physica A Statistical Mechanics and ItsApplications vol 429 pp 169ndash176 2015

[20] Y Yu ldquoNode importance measurement based on the degreeand closeness centralityrdquo Journal of Information and Com-putational Science vol 12 no 3 pp 1281ndash1291 2015

12 Complexity

[21] Q Hu Y Yin P Ma Y Gao Y Zhang and C Xing ldquoA newapproach to identify influential spreaders in complex net-worksrdquoActa Physica Sinica vol 62 no 14 (in Chinese) 2013

[22] Y Zhang Y Bao S Zhao J Chen and J Tang ldquoIdentifyingnode importance by combining betweenness centrality andKatz centralityrdquo in Proceedings of the 2015 InternationalConference on Cloud Computing and Big Data (Ccbd)pp 354ndash357 (in English) Shanghai China November 2015

[23] Y Yang and G Xie ldquoEfficient identification of node im-portance in social networksrdquo Information Processing ampManagement vol 52 no 5 pp 911ndash922 2016 (in English)

[24] Z Liu C Jiang J Wang and H Yu ldquo+e node importance inactual complex networks based on a multi-attribute rankingmethodrdquo Knowledge-Based Systems vol 84 pp 56ndash66 2015(in English)

[25] C E Shannon ldquoA mathematical theory of communicationrdquoBell System Technical Journal vol 27 no 4 pp 623ndash656 1948

[26] Q Zhang M Li and Y Deng ldquoMeasure the structure sim-ilarity of nodes in complex networks based on relative en-tropyrdquo Physica A Statistical Mechanics and Its Applicationsvol 491 pp 749ndash763 2018 (in English)

[27] Y He H Guo M Jin and P Ren ldquoA linguistic entropyweight method and its application in linguistic multi-attributegroup decision makingrdquo Nonlinear Dynamics vol 84 no 1pp 399ndash404 2016 (in English)

[28] X Liang and C Wei ldquoAn Atanassovrsquos intuitionistic fuzzymulti-attribute group decision making method based onentropy and similarity measurerdquo International Journal ofMachine Learning and Cybernetics vol 5 no 3 pp 435ndash4442014 (in English)

[29] T-Y Chen and C-H Li ldquoObjective weights with intuition-istic fuzzy entropy measures and computational experimentanalysisrdquo Applied Soft Computing vol 11 no 8 pp 5411ndash5423 2011 (in English)

[30] S-P Wan Q-Y Wang and J-Y Dong ldquo+e extendedVIKOR method for multi-attribute group decision makingwith triangular intuitionistic fuzzy numbersrdquo Knowledge-Based Systems vol 52 pp 65ndash77 2013 (in English)

[31] H Nguyen ldquoA new knowledge-based measure for intui-tionistic fuzzy sets and its application in multiple attributegroup decision makingrdquo Expert Systems with Applicationsvol 42 no 22 pp 8766ndash8774 2015 (in English)

[32] S Dereich and P Morters ldquoRandom networks with sublinearpreferential attachment the giant componentrdquoampe Annals ofProbability vol 41 no 1 pp 329ndash384 2013 (in English)

[33] C M Schneider A A Moreira J S Andrade S Havlin andH J Herrmann ldquoMitigation of malicious attacks on net-worksrdquo Proceedings of the National Academy of Sciences of theUnited States of America vol 108 no 10 pp 3838ndash3841 2011(in English)

[34] A Zareie and A Sheikhahmadi ldquoA hierarchical approach forinfluential node ranking in complex social networksrdquo ExpertSystems with Applications vol 93 pp 200ndash211 2018 (inEnglish)

Complexity 13

44 Node Importance Ranking +e node importance iscalculated by the following

si 11139444

j1wjbij (11)

+e larger si(i 1 2 n) is the more important thenode is

+e general node importance calculation and noderanking steps in complex networks is shown in Algorithm 1Firstly determine the indicators DCCCBC IKs andcalculate the value of the four indicators for all nodes in thecomplex network +en we construct the attribute matrix P

based on equation (5) +irdly we calculate the normalizedattribute matrix R based on equations (6) and (7) Fourthlywe calculate the entropy e of each indicator based on

Ks = 1Ks = 2Ks = 3

(a)

IKs = 1IKs = 2IKs = 3

IKs = 4IKs = 5IKs = 6

(b)

Figure 1 (a) the schematic diagram of K-shell decomposition (b) the schematic diagram of improved K-shell decomposition

11

9

19

10

187

8

12

6

20

2624

29252327

28 30

22

36

35

34

21 31

32 33

37

14

13

16

15

17

2

1

3

4

5

Figure 2 Topological structure of BUPT campus network

Complexity 5

equations (8) and (9) (lines 1ndash11) +en the weights of allindicators could be obtained based on equation (10) (lines14ndash16) and the node importance is calculated based onequation (11) (lines 17ndash19) Finally all nodes are rankedbased on the node importance (lines 20ndash21) +e head of thenode list is the most importance node +e time complexityof our algorithm is O(n) where n denotes the number ofnodes in the network

5 Experiments

We conducted the experiments on three real networks andcompared the results of our method with the random se-lection method (Random) and the TOPSIS-RE method in

[24] +e experimental result proves that our method per-forms better

51 Experiment Setup +e selected networks are the campusnetwork of Beijing University of Posts and Telecommunications(BUPT) Shanxi Water Network and Shanxi Railway NetworkFirst we prove the effectiveness of ourmethod by experimentingon the BUPT campus network +en we illustrate the experi-mental results on Shanxi Water Network and Shanxi RailwayNetwork to see how the proposed method works in morecomplicated cases +e experiment is conducted on a PC withIntel Core i5-3470 32GHz CPU 4GB RAM

TOPSIS-RE extensively employs the Technique forOrder Preference by Similarity to Ideal Object (TOPSIS) to

00

02

04

06

08

10

12

DC CC BCCentrality measure

IKsEn

trop

y an

d w

eigh

t

EntropyWeight

Figure 3 +e relationship between entropy and weight

Input the normalized attribute matrix R

Output the ranking result(1) for each Ij in I do(2) sumj 0(3) for each vi in V do(4) sumj sumj + bij(5) end(6) ensumj 0(7) for each vi in V do(8) pij bijsumj(9) ensumj ensumj minus pij lnpij(10) end(11) ej ensumjln n

(12) esum esum + ej

(13) end(14) for each Ij in I do(15) wj 1 minus ej4 minus esum(16) end(17) for each vi in V do(18) si 1113936

4j1 wjbij

(19) end(20) Rank the node list based on si

(21) return the ranked node list

ALGORITHM 1 Node importance ranking algorithm

6 Complexity

evaluate the node importance +e core idea of TOPSIS-REis to construct a positive ideal object and a negative idealobject from the original data +e positive ideal object iscalculated based on the max value of the indicators and thenegative ideal object is calculated based on min value of the

indicators All methods are implemented by using thenetwork analysis software Cytoscape together with Javaprogramming language

In the experiments all nodes are ranked based on the nodeimportance+en the nodes are removed one by one from the

Table 1 Ranking result of BUPT campus network

Method Ranking resultsEBSAM 8 31 11 3 18 22 7 20 21 4 9 10 14 24 26 28TOPSIS-RE 8 18 31 3 11 7 21 20 9 10 4 14 24 26 28 22Random 19 12 23 25 30 32 13 34 16 5 27 21 9 20 6 10

6

18

9

19

20

10

12

7

282624

2725 2923

30

33

37

22

21

32

3435

36

13

16

15

14

17

1

4

2

5

(a)

6

109

19

12 282624

2725 2923

30

33

37

36

21

32

3435

13

16

15

14

17

1

4

2

5

(b)

6

109

19

121

5

2

282624

2725 2923

30

33

37

36

21

32

3435

13

16

15

14

17

(c)

65

2

19

121

2923

30

2725

33

37

36

21

32

3435

13

16

15

17

(d)

Figure 4 +e removing process with EBSAM on BUPTcampus network (a) 4 nodes have been removed (b) 8 nodes have been removed(c) 12 nodes have been removed (d) 16 nodes have been removed

02468

101214161820

1

NCC

3 5 7 9 11 13 15 17e number of nodes to be attacked

19 21 23 25 27 29 31 33 35 37

TOPSIS-REEBSAMRandom

Figure 5 Experiment results on BUPT campus network

Complexity 7

networks according to the ranking results +e Number ofConnected Components (NCC) is employed to evaluate theeffectiveness of the methods A connected component of anundirected network is a subgraph in which any two nodes areconnected to each other by edges After we remove one ormore nodes in a network the network will be divided intoseveral disconnected subgraphs Any node inside a subgraph isreachable from other nodes in the same subgraph +ere is nopath between two nodes belonging to different a subgraphNCC is the number of these disconnected subgraphs NCCreflects the connectivity of a network +e robustness of anetwork could be measured by calculating the size of thelargest connected component after removing a fraction of thenodes [32ndash34] +e number of connected component in anetwork could reflect its connectivity A larger value of NCCreflects that the network is divided into more disconnected

subgraphs which indicates the node you remove is moreimportant respect to network connectivity +erefore a largervalue of NCC indicates a better performance A node isconsidered to bemore important ifmore number of connectedcomponent increases after it has been removed

52 Experimental Results

521 Experiment Results on BUPT Campus Network+e topological structure of BUPT campus network is il-lustrated in Figure 2 +e number in each node is just anidentity of the node It does not have any meaning except toidentify different nodes +e node can be identified by itsnumber in the graph

+e node importance rank results of EBSAM TOPSIS-RE and a random selection algorithm (Random) are

Yun Cheng

Jin ChengLin Fen

Chang Zhi

Jin Zhong

Tai Yuan Yang Quan

Qi Zhou

Su Zhou

Da Tong

Figure 6 Shanxi water network

8 Complexity

0

5

10

15

20

25

30

35

1 3 5 7

NCC

9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43 45 47 49 51 53 55 57 59 61 63 65 67 69 71 73 75 77 79 81

TOPSIS-REEBSAMRandom

The number of nodes to be attacked

Figure 8 NCC of Shanxi water network

75

71

48

7473

81

68

8078

77

76

82

79

72

5655

70

69

57

54

51

5253

5049

44

33

4332

46

45

59

61 65

63

67

47

60

6664

62

58

16

35

27

14

28

34

221513

24 26

25

31

2021

10

12

2

9

1

38

36

29

39

1719

23

18

6

78

11

5

4

3

42

41

37

40

30

Figure 7 Topological structure of Shanxi water network

Complexity 9

illustrated in Table 1 We only list the top 16 nodes in therank results because the rank results of the rest nodes are thesame in EBSAM and TOPSIS-RE

According to the rank result we remove the nodes oneby one from the network until all nodes have been re-moved from the network We calculate the number ofconnected components in the network after removing anode +e removing process of EBSAM is shown in Fig-ure 4 We list out the topological structure of the networkafter removing every four nodes +e NCC of EBSAM andTOPSIS-RE and Random methods are shown in Figure 5As we can see the number of connected components ofRandom method is much less than the other two methods+erefore Random is less effective in destroying thenetwork by attacking the important nodes We can also seethat the number of connected components of EBSAM ismore than TOPSIS-RE Hence the connectivity of thenetwork is worse with EBSAM Attacking the network

based on the ranking result of EBSAM is more effectivethan TOPSIS-RE +at is because we obtain the weight ofthe four indicators objectively and adaptively other thanassign a fixed value subjectively

522 Experimental Results on Shanxi Water NetworkAs shown in Figure 6 Shanxi Water Network plays a vitalrole in the normal production and living activities+e greenline in Figure 6 denotes the water supply network +eShanxi water network provides guarantee for water demandof north China and its topological structure is shown inFigure 7 As shown in Figure 7 Shanxi Water Network iscomposed of 82 nodes +e experimental result is shown inFigure 8 As we can see the connectivity of the network hasbeen destroyed after the top 50 nodes have been attackedHowever NCC of our method is larger than the other twocompared methods +erefore the performance of EBSAMis better than other compared methods

Da Tong

Su Zhou

Qi Zhou

Tai Yuan Yang Quan

Jin Zhong

Lin Fen

Chang Zhi

Jin Cheng

Yun Cheng

The first 1000kmThe second 1000kmThe third 1000kmExpressway under construction and to be built

Figure 9 Shanxi railway network

10 Complexity

523 Experimental Results on Shanxi Railway NetworkFinally we conduct experiment on Shanxi Railway Net-work As shown in Figure 9 Shanxi Railway Network is apart of the transportation network in Shanxi It providesgreat convenience for peoplersquos outgoing and commodities

trading +e topological structure of Shanxi Water Net-work is shown in Figure 10 +e experimental result isshown in Figure 11 +e network is coming to break downafter the top 60 nodes have been attacked +e NCC ofShanxi Water Network obtains the largest ascent with our

4

31811

1

2

13

1219

2123

22

20

9

24

31

6

25

10

87

5

26

14

28

16 17

15

29

93

85

90

96

91

92

84

94

9597

98

33

27

30

3435

43

5150

3236

45

44

58

4138

40

3937

42

81

6465 66

7982

80 78

76

77

83

53

52

59

54

60

55

61

4948

46

47

86

62

87

63

57

89

56

886968

71

73 72

70

7475

67

Figure 10 Topological structure of Shanxi railway network

0

5

10

15

20

25

30

35

40

1 5 9 13 17

NCC

21 25 29 33 37 41 45 49 53 57 61 65 69 73 77 81 85 89 93 97

TOPSIS-REEBSAMRandom

The number of nodes to be attacked

Figure 11 NCC of Shanxi railway network

Complexity 11

method +erefore EBSAM performs better than othermethods

6 Conclusions and Future Work

In this paper we proposed an entropy theory-based self-adaptive node importance evaluation method for complexnetworks Firstly we select four centrality measures whichcan reflect different characteristics of the node as nodeimportance evaluation indicators +en we combine themtogether with appropriate weights calculated by an entropytheory-based algorithm +e algorithm shows a strongadaptability and thus allows be widely implemented indifferent kinds of networks In the traditional method theweights are selected based on the subjective experience of theresearchers instead of enough scientific basis which wouldlead to inaccurate evaluation results +e proposed methodis better because it utilizes entropy theory to calculate theweight of each indicator A smaller value of entropy indicatesthat the corresponding attribution contains less useful in-formation In a multiattribute decision-making problem weneed to assign a larger weight to attribute with more usefulinformation rather than the attribute with greater uncer-tainty So with this algorithm we can better assign properweight to different attributions +e experimental results onthree types of real-world complex networks show that ourmethod performs better with compared methods Our on-going research will focus on investigating the effectiveness ofour method in more complex environments

Data Availability

+e data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

+e authors declare that they have no conflicts of interest

Acknowledgments

+is study was supported by National Key RampD Program ofChina (Funding No 2018YFB1402800) and the NaturalScience Foundation of China (No 61571066)

References

[1] SWang X Zhang Y Zhang LWang J Yang andWWangldquoA survey on mobile edge networks convergence of com-puting caching and communicationsrdquo Ieee Access vol 5pp 6757ndash6779 2017 (in English)

[2] X Ren and L Lu ldquoReview of ranking nodes in complexnetworksrdquo Chinese Science Bulletin vol 59 no 13pp 1175ndash1197 2014

[3] K L S Samanta and M Pal ldquoStudy on centrality measures insocial networks a surveyrdquo Social Network Analysis Miningvol 8 no 1 p 13 2018

[4] J Dong F Ye W Chen and J Wu ldquoIdentifying influentialnodes in complex networks via semi-local centralityrdquo inProceedings of the 2018 IEEE International Symposium on

Circuits and Systems (Iscas) (in English) Florence Italy May2018

[5] K Angelou M Maragakis and P Argyrakis ldquoA structuralanalysis of the patent citation network by the k-shell de-composition methodrdquo Physica A-Statistical Mechanics and ItsApplications vol 521 pp 476ndash483 2019 (in English)

[6] G Li M Li J Wang Y Li and Y Pan ldquoUnited neighborhoodcloseness centrality and orthology for predicting essentialproteinsrdquo IEEEACM Transactions on Computational Biologyand Bioinformatics 2018

[7] M Riondato and E Upfal ldquoABRA approximating be-tweenness centrality in static and dynamic graphs withrademacher averagesrdquo ACM Transactions on KnowledgeDiscovery from Data vol 12 no 5 pp 1ndash38 2018 (inEnglish)

[8] N Gupta A Singh and H Cherifi ldquoCentrality measures fornetworks with community structurerdquo Physica A StatisticalMechanics and Its Applications vol 452 pp 46ndash59 2016 (inEnglish)

[9] D Chen H Gao L Lu and T Zhou ldquoIdentifying influentialnodes in large-scale directed networks the role of clusteringrdquoPLoS One vol 8 Article ID e77455 no 10 (in English) 2013

[10] K Tu P Cui X Wang P Yu and W Zhu ldquoDeep recursivenetwork embedding with regular equivalencerdquo in Proceedingsof the Kddrsquo18 24th Acm Sigkdd International Conference onKnowledge Discovery amp Data Mining London UKpp 2357ndash2366 August 2018 (in English)

[11] R Poulin M-C Boily and B R Masse ldquoDynamical systemsto define centrality in social networksrdquo Social Networksvol 22 no 3 pp 187ndash220 2000 (in English)

[12] P Hu and T Mei ldquoRanking influential nodes in complexnetworks with structural holesrdquo Physica A Statistical Me-chanics and Its Applications vol 490 pp 624ndash631 15 2018 (inEnglish)

[13] L Lu Y Zhang C Yeung and T Zhou ldquoLeaders in socialnetworks the delicious caserdquo Plos One vol 6 no 6 (inEnglish) 2011

[14] J Liu Q Xiong W Shi X Shi and K Wang ldquoEvaluating theimportance of nodes in complex networksrdquo Physica A Sta-tistical Mechanics and Its Applications vol 452 pp 209ndash2192016 (in English)

[15] F Hu Y Liu and J Jin ldquoMulti-index evaluation algorithmbased on locally linear embedding for the node importance incomplex networksrdquo in Proceedings of ampirteenth InternationalSymposium on Distributed Computing and Applications toBusiness Engineering and Science (Dcabes 2014) pp 138ndash142(in English) Xian Ning China November 2014

[16] S T Roweis and L K Saul ldquoNonlinear dimensionality re-duction by locally linear embeddingrdquo Science vol 290no 5500 pp 2323ndash2326 2000 (in English)

[17] L Xu J Liu Y Liu Y Liu J Gou and B Masoud ldquoNodeimportance classified comprehensive assessmentrdquo Proceed-ings of the Csee vol 34 no 10 pp 1609ndash1617 2014

[18] X Zhang and X Wang ldquoEvaluation formula for communi-cation network node importancerdquo Journal of NortheasternUniversity vol 35 no 5 pp 663ndash666 2014

[19] H Ping W Fan and S Mei ldquoIdentifying node importance incomplex networksrdquo Physica A Statistical Mechanics and ItsApplications vol 429 pp 169ndash176 2015

[20] Y Yu ldquoNode importance measurement based on the degreeand closeness centralityrdquo Journal of Information and Com-putational Science vol 12 no 3 pp 1281ndash1291 2015

12 Complexity

[21] Q Hu Y Yin P Ma Y Gao Y Zhang and C Xing ldquoA newapproach to identify influential spreaders in complex net-worksrdquoActa Physica Sinica vol 62 no 14 (in Chinese) 2013

[22] Y Zhang Y Bao S Zhao J Chen and J Tang ldquoIdentifyingnode importance by combining betweenness centrality andKatz centralityrdquo in Proceedings of the 2015 InternationalConference on Cloud Computing and Big Data (Ccbd)pp 354ndash357 (in English) Shanghai China November 2015

[23] Y Yang and G Xie ldquoEfficient identification of node im-portance in social networksrdquo Information Processing ampManagement vol 52 no 5 pp 911ndash922 2016 (in English)

[24] Z Liu C Jiang J Wang and H Yu ldquo+e node importance inactual complex networks based on a multi-attribute rankingmethodrdquo Knowledge-Based Systems vol 84 pp 56ndash66 2015(in English)

[25] C E Shannon ldquoA mathematical theory of communicationrdquoBell System Technical Journal vol 27 no 4 pp 623ndash656 1948

[26] Q Zhang M Li and Y Deng ldquoMeasure the structure sim-ilarity of nodes in complex networks based on relative en-tropyrdquo Physica A Statistical Mechanics and Its Applicationsvol 491 pp 749ndash763 2018 (in English)

[27] Y He H Guo M Jin and P Ren ldquoA linguistic entropyweight method and its application in linguistic multi-attributegroup decision makingrdquo Nonlinear Dynamics vol 84 no 1pp 399ndash404 2016 (in English)

[28] X Liang and C Wei ldquoAn Atanassovrsquos intuitionistic fuzzymulti-attribute group decision making method based onentropy and similarity measurerdquo International Journal ofMachine Learning and Cybernetics vol 5 no 3 pp 435ndash4442014 (in English)

[29] T-Y Chen and C-H Li ldquoObjective weights with intuition-istic fuzzy entropy measures and computational experimentanalysisrdquo Applied Soft Computing vol 11 no 8 pp 5411ndash5423 2011 (in English)

[30] S-P Wan Q-Y Wang and J-Y Dong ldquo+e extendedVIKOR method for multi-attribute group decision makingwith triangular intuitionistic fuzzy numbersrdquo Knowledge-Based Systems vol 52 pp 65ndash77 2013 (in English)

[31] H Nguyen ldquoA new knowledge-based measure for intui-tionistic fuzzy sets and its application in multiple attributegroup decision makingrdquo Expert Systems with Applicationsvol 42 no 22 pp 8766ndash8774 2015 (in English)

[32] S Dereich and P Morters ldquoRandom networks with sublinearpreferential attachment the giant componentrdquoampe Annals ofProbability vol 41 no 1 pp 329ndash384 2013 (in English)

[33] C M Schneider A A Moreira J S Andrade S Havlin andH J Herrmann ldquoMitigation of malicious attacks on net-worksrdquo Proceedings of the National Academy of Sciences of theUnited States of America vol 108 no 10 pp 3838ndash3841 2011(in English)

[34] A Zareie and A Sheikhahmadi ldquoA hierarchical approach forinfluential node ranking in complex social networksrdquo ExpertSystems with Applications vol 93 pp 200ndash211 2018 (inEnglish)

Complexity 13

equations (8) and (9) (lines 1ndash11) +en the weights of allindicators could be obtained based on equation (10) (lines14ndash16) and the node importance is calculated based onequation (11) (lines 17ndash19) Finally all nodes are rankedbased on the node importance (lines 20ndash21) +e head of thenode list is the most importance node +e time complexityof our algorithm is O(n) where n denotes the number ofnodes in the network

5 Experiments

We conducted the experiments on three real networks andcompared the results of our method with the random se-lection method (Random) and the TOPSIS-RE method in

[24] +e experimental result proves that our method per-forms better

51 Experiment Setup +e selected networks are the campusnetwork of Beijing University of Posts and Telecommunications(BUPT) Shanxi Water Network and Shanxi Railway NetworkFirst we prove the effectiveness of ourmethod by experimentingon the BUPT campus network +en we illustrate the experi-mental results on Shanxi Water Network and Shanxi RailwayNetwork to see how the proposed method works in morecomplicated cases +e experiment is conducted on a PC withIntel Core i5-3470 32GHz CPU 4GB RAM

TOPSIS-RE extensively employs the Technique forOrder Preference by Similarity to Ideal Object (TOPSIS) to

00

02

04

06

08

10

12

DC CC BCCentrality measure

IKsEn

trop

y an

d w

eigh

t

EntropyWeight

Figure 3 +e relationship between entropy and weight

Input the normalized attribute matrix R

Output the ranking result(1) for each Ij in I do(2) sumj 0(3) for each vi in V do(4) sumj sumj + bij(5) end(6) ensumj 0(7) for each vi in V do(8) pij bijsumj(9) ensumj ensumj minus pij lnpij(10) end(11) ej ensumjln n

(12) esum esum + ej

(13) end(14) for each Ij in I do(15) wj 1 minus ej4 minus esum(16) end(17) for each vi in V do(18) si 1113936

4j1 wjbij

(19) end(20) Rank the node list based on si

(21) return the ranked node list

ALGORITHM 1 Node importance ranking algorithm

6 Complexity

evaluate the node importance +e core idea of TOPSIS-REis to construct a positive ideal object and a negative idealobject from the original data +e positive ideal object iscalculated based on the max value of the indicators and thenegative ideal object is calculated based on min value of the

indicators All methods are implemented by using thenetwork analysis software Cytoscape together with Javaprogramming language

In the experiments all nodes are ranked based on the nodeimportance+en the nodes are removed one by one from the

Table 1 Ranking result of BUPT campus network

Method Ranking resultsEBSAM 8 31 11 3 18 22 7 20 21 4 9 10 14 24 26 28TOPSIS-RE 8 18 31 3 11 7 21 20 9 10 4 14 24 26 28 22Random 19 12 23 25 30 32 13 34 16 5 27 21 9 20 6 10

6

18

9

19

20

10

12

7

282624

2725 2923

30

33

37

22

21

32

3435

36

13

16

15

14

17

1

4

2

5

(a)

6

109

19

12 282624

2725 2923

30

33

37

36

21

32

3435

13

16

15

14

17

1

4

2

5

(b)

6

109

19

121

5

2

282624

2725 2923

30

33

37

36

21

32

3435

13

16

15

14

17

(c)

65

2

19

121

2923

30

2725

33

37

36

21

32

3435

13

16

15

17

(d)

Figure 4 +e removing process with EBSAM on BUPTcampus network (a) 4 nodes have been removed (b) 8 nodes have been removed(c) 12 nodes have been removed (d) 16 nodes have been removed

02468

101214161820

1

NCC

3 5 7 9 11 13 15 17e number of nodes to be attacked

19 21 23 25 27 29 31 33 35 37

TOPSIS-REEBSAMRandom

Figure 5 Experiment results on BUPT campus network

Complexity 7

networks according to the ranking results +e Number ofConnected Components (NCC) is employed to evaluate theeffectiveness of the methods A connected component of anundirected network is a subgraph in which any two nodes areconnected to each other by edges After we remove one ormore nodes in a network the network will be divided intoseveral disconnected subgraphs Any node inside a subgraph isreachable from other nodes in the same subgraph +ere is nopath between two nodes belonging to different a subgraphNCC is the number of these disconnected subgraphs NCCreflects the connectivity of a network +e robustness of anetwork could be measured by calculating the size of thelargest connected component after removing a fraction of thenodes [32ndash34] +e number of connected component in anetwork could reflect its connectivity A larger value of NCCreflects that the network is divided into more disconnected

subgraphs which indicates the node you remove is moreimportant respect to network connectivity +erefore a largervalue of NCC indicates a better performance A node isconsidered to bemore important ifmore number of connectedcomponent increases after it has been removed

52 Experimental Results

521 Experiment Results on BUPT Campus Network+e topological structure of BUPT campus network is il-lustrated in Figure 2 +e number in each node is just anidentity of the node It does not have any meaning except toidentify different nodes +e node can be identified by itsnumber in the graph

+e node importance rank results of EBSAM TOPSIS-RE and a random selection algorithm (Random) are

Yun Cheng

Jin ChengLin Fen

Chang Zhi

Jin Zhong

Tai Yuan Yang Quan

Qi Zhou

Su Zhou

Da Tong

Figure 6 Shanxi water network

8 Complexity

0

5

10

15

20

25

30

35

1 3 5 7

NCC

9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43 45 47 49 51 53 55 57 59 61 63 65 67 69 71 73 75 77 79 81

TOPSIS-REEBSAMRandom

The number of nodes to be attacked

Figure 8 NCC of Shanxi water network

75

71

48

7473

81

68

8078

77

76

82

79

72

5655

70

69

57

54

51

5253

5049

44

33

4332

46

45

59

61 65

63

67

47

60

6664

62

58

16

35

27

14

28

34

221513

24 26

25

31

2021

10

12

2

9

1

38

36

29

39

1719

23

18

6

78

11

5

4

3

42

41

37

40

30

Figure 7 Topological structure of Shanxi water network

Complexity 9

illustrated in Table 1 We only list the top 16 nodes in therank results because the rank results of the rest nodes are thesame in EBSAM and TOPSIS-RE

According to the rank result we remove the nodes oneby one from the network until all nodes have been re-moved from the network We calculate the number ofconnected components in the network after removing anode +e removing process of EBSAM is shown in Fig-ure 4 We list out the topological structure of the networkafter removing every four nodes +e NCC of EBSAM andTOPSIS-RE and Random methods are shown in Figure 5As we can see the number of connected components ofRandom method is much less than the other two methods+erefore Random is less effective in destroying thenetwork by attacking the important nodes We can also seethat the number of connected components of EBSAM ismore than TOPSIS-RE Hence the connectivity of thenetwork is worse with EBSAM Attacking the network

based on the ranking result of EBSAM is more effectivethan TOPSIS-RE +at is because we obtain the weight ofthe four indicators objectively and adaptively other thanassign a fixed value subjectively

522 Experimental Results on Shanxi Water NetworkAs shown in Figure 6 Shanxi Water Network plays a vitalrole in the normal production and living activities+e greenline in Figure 6 denotes the water supply network +eShanxi water network provides guarantee for water demandof north China and its topological structure is shown inFigure 7 As shown in Figure 7 Shanxi Water Network iscomposed of 82 nodes +e experimental result is shown inFigure 8 As we can see the connectivity of the network hasbeen destroyed after the top 50 nodes have been attackedHowever NCC of our method is larger than the other twocompared methods +erefore the performance of EBSAMis better than other compared methods

Da Tong

Su Zhou

Qi Zhou

Tai Yuan Yang Quan

Jin Zhong

Lin Fen

Chang Zhi

Jin Cheng

Yun Cheng

The first 1000kmThe second 1000kmThe third 1000kmExpressway under construction and to be built

Figure 9 Shanxi railway network

10 Complexity

523 Experimental Results on Shanxi Railway NetworkFinally we conduct experiment on Shanxi Railway Net-work As shown in Figure 9 Shanxi Railway Network is apart of the transportation network in Shanxi It providesgreat convenience for peoplersquos outgoing and commodities

trading +e topological structure of Shanxi Water Net-work is shown in Figure 10 +e experimental result isshown in Figure 11 +e network is coming to break downafter the top 60 nodes have been attacked +e NCC ofShanxi Water Network obtains the largest ascent with our

4

31811

1

2

13

1219

2123

22

20

9

24

31

6

25

10

87

5

26

14

28

16 17

15

29

93

85

90

96

91

92

84

94

9597

98

33

27

30

3435

43

5150

3236

45

44

58

4138

40

3937

42

81

6465 66

7982

80 78

76

77

83

53

52

59

54

60

55

61

4948

46

47

86

62

87

63

57

89

56

886968

71

73 72

70

7475

67

Figure 10 Topological structure of Shanxi railway network

0

5

10

15

20

25

30

35

40

1 5 9 13 17

NCC

21 25 29 33 37 41 45 49 53 57 61 65 69 73 77 81 85 89 93 97

TOPSIS-REEBSAMRandom

The number of nodes to be attacked

Figure 11 NCC of Shanxi railway network

Complexity 11

method +erefore EBSAM performs better than othermethods

6 Conclusions and Future Work

In this paper we proposed an entropy theory-based self-adaptive node importance evaluation method for complexnetworks Firstly we select four centrality measures whichcan reflect different characteristics of the node as nodeimportance evaluation indicators +en we combine themtogether with appropriate weights calculated by an entropytheory-based algorithm +e algorithm shows a strongadaptability and thus allows be widely implemented indifferent kinds of networks In the traditional method theweights are selected based on the subjective experience of theresearchers instead of enough scientific basis which wouldlead to inaccurate evaluation results +e proposed methodis better because it utilizes entropy theory to calculate theweight of each indicator A smaller value of entropy indicatesthat the corresponding attribution contains less useful in-formation In a multiattribute decision-making problem weneed to assign a larger weight to attribute with more usefulinformation rather than the attribute with greater uncer-tainty So with this algorithm we can better assign properweight to different attributions +e experimental results onthree types of real-world complex networks show that ourmethod performs better with compared methods Our on-going research will focus on investigating the effectiveness ofour method in more complex environments

Data Availability

+e data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

+e authors declare that they have no conflicts of interest

Acknowledgments

+is study was supported by National Key RampD Program ofChina (Funding No 2018YFB1402800) and the NaturalScience Foundation of China (No 61571066)

References

[1] SWang X Zhang Y Zhang LWang J Yang andWWangldquoA survey on mobile edge networks convergence of com-puting caching and communicationsrdquo Ieee Access vol 5pp 6757ndash6779 2017 (in English)

[2] X Ren and L Lu ldquoReview of ranking nodes in complexnetworksrdquo Chinese Science Bulletin vol 59 no 13pp 1175ndash1197 2014

[3] K L S Samanta and M Pal ldquoStudy on centrality measures insocial networks a surveyrdquo Social Network Analysis Miningvol 8 no 1 p 13 2018

[4] J Dong F Ye W Chen and J Wu ldquoIdentifying influentialnodes in complex networks via semi-local centralityrdquo inProceedings of the 2018 IEEE International Symposium on

Circuits and Systems (Iscas) (in English) Florence Italy May2018

[5] K Angelou M Maragakis and P Argyrakis ldquoA structuralanalysis of the patent citation network by the k-shell de-composition methodrdquo Physica A-Statistical Mechanics and ItsApplications vol 521 pp 476ndash483 2019 (in English)

[6] G Li M Li J Wang Y Li and Y Pan ldquoUnited neighborhoodcloseness centrality and orthology for predicting essentialproteinsrdquo IEEEACM Transactions on Computational Biologyand Bioinformatics 2018

[7] M Riondato and E Upfal ldquoABRA approximating be-tweenness centrality in static and dynamic graphs withrademacher averagesrdquo ACM Transactions on KnowledgeDiscovery from Data vol 12 no 5 pp 1ndash38 2018 (inEnglish)

[8] N Gupta A Singh and H Cherifi ldquoCentrality measures fornetworks with community structurerdquo Physica A StatisticalMechanics and Its Applications vol 452 pp 46ndash59 2016 (inEnglish)

[9] D Chen H Gao L Lu and T Zhou ldquoIdentifying influentialnodes in large-scale directed networks the role of clusteringrdquoPLoS One vol 8 Article ID e77455 no 10 (in English) 2013

[10] K Tu P Cui X Wang P Yu and W Zhu ldquoDeep recursivenetwork embedding with regular equivalencerdquo in Proceedingsof the Kddrsquo18 24th Acm Sigkdd International Conference onKnowledge Discovery amp Data Mining London UKpp 2357ndash2366 August 2018 (in English)

[11] R Poulin M-C Boily and B R Masse ldquoDynamical systemsto define centrality in social networksrdquo Social Networksvol 22 no 3 pp 187ndash220 2000 (in English)

[12] P Hu and T Mei ldquoRanking influential nodes in complexnetworks with structural holesrdquo Physica A Statistical Me-chanics and Its Applications vol 490 pp 624ndash631 15 2018 (inEnglish)

[13] L Lu Y Zhang C Yeung and T Zhou ldquoLeaders in socialnetworks the delicious caserdquo Plos One vol 6 no 6 (inEnglish) 2011

[14] J Liu Q Xiong W Shi X Shi and K Wang ldquoEvaluating theimportance of nodes in complex networksrdquo Physica A Sta-tistical Mechanics and Its Applications vol 452 pp 209ndash2192016 (in English)

[15] F Hu Y Liu and J Jin ldquoMulti-index evaluation algorithmbased on locally linear embedding for the node importance incomplex networksrdquo in Proceedings of ampirteenth InternationalSymposium on Distributed Computing and Applications toBusiness Engineering and Science (Dcabes 2014) pp 138ndash142(in English) Xian Ning China November 2014

[16] S T Roweis and L K Saul ldquoNonlinear dimensionality re-duction by locally linear embeddingrdquo Science vol 290no 5500 pp 2323ndash2326 2000 (in English)

[17] L Xu J Liu Y Liu Y Liu J Gou and B Masoud ldquoNodeimportance classified comprehensive assessmentrdquo Proceed-ings of the Csee vol 34 no 10 pp 1609ndash1617 2014

[18] X Zhang and X Wang ldquoEvaluation formula for communi-cation network node importancerdquo Journal of NortheasternUniversity vol 35 no 5 pp 663ndash666 2014

[19] H Ping W Fan and S Mei ldquoIdentifying node importance incomplex networksrdquo Physica A Statistical Mechanics and ItsApplications vol 429 pp 169ndash176 2015

[20] Y Yu ldquoNode importance measurement based on the degreeand closeness centralityrdquo Journal of Information and Com-putational Science vol 12 no 3 pp 1281ndash1291 2015

12 Complexity

[21] Q Hu Y Yin P Ma Y Gao Y Zhang and C Xing ldquoA newapproach to identify influential spreaders in complex net-worksrdquoActa Physica Sinica vol 62 no 14 (in Chinese) 2013

[22] Y Zhang Y Bao S Zhao J Chen and J Tang ldquoIdentifyingnode importance by combining betweenness centrality andKatz centralityrdquo in Proceedings of the 2015 InternationalConference on Cloud Computing and Big Data (Ccbd)pp 354ndash357 (in English) Shanghai China November 2015

[23] Y Yang and G Xie ldquoEfficient identification of node im-portance in social networksrdquo Information Processing ampManagement vol 52 no 5 pp 911ndash922 2016 (in English)

[24] Z Liu C Jiang J Wang and H Yu ldquo+e node importance inactual complex networks based on a multi-attribute rankingmethodrdquo Knowledge-Based Systems vol 84 pp 56ndash66 2015(in English)

[25] C E Shannon ldquoA mathematical theory of communicationrdquoBell System Technical Journal vol 27 no 4 pp 623ndash656 1948

[26] Q Zhang M Li and Y Deng ldquoMeasure the structure sim-ilarity of nodes in complex networks based on relative en-tropyrdquo Physica A Statistical Mechanics and Its Applicationsvol 491 pp 749ndash763 2018 (in English)

[27] Y He H Guo M Jin and P Ren ldquoA linguistic entropyweight method and its application in linguistic multi-attributegroup decision makingrdquo Nonlinear Dynamics vol 84 no 1pp 399ndash404 2016 (in English)

[28] X Liang and C Wei ldquoAn Atanassovrsquos intuitionistic fuzzymulti-attribute group decision making method based onentropy and similarity measurerdquo International Journal ofMachine Learning and Cybernetics vol 5 no 3 pp 435ndash4442014 (in English)

[29] T-Y Chen and C-H Li ldquoObjective weights with intuition-istic fuzzy entropy measures and computational experimentanalysisrdquo Applied Soft Computing vol 11 no 8 pp 5411ndash5423 2011 (in English)

[30] S-P Wan Q-Y Wang and J-Y Dong ldquo+e extendedVIKOR method for multi-attribute group decision makingwith triangular intuitionistic fuzzy numbersrdquo Knowledge-Based Systems vol 52 pp 65ndash77 2013 (in English)

[31] H Nguyen ldquoA new knowledge-based measure for intui-tionistic fuzzy sets and its application in multiple attributegroup decision makingrdquo Expert Systems with Applicationsvol 42 no 22 pp 8766ndash8774 2015 (in English)

[32] S Dereich and P Morters ldquoRandom networks with sublinearpreferential attachment the giant componentrdquoampe Annals ofProbability vol 41 no 1 pp 329ndash384 2013 (in English)

[33] C M Schneider A A Moreira J S Andrade S Havlin andH J Herrmann ldquoMitigation of malicious attacks on net-worksrdquo Proceedings of the National Academy of Sciences of theUnited States of America vol 108 no 10 pp 3838ndash3841 2011(in English)

[34] A Zareie and A Sheikhahmadi ldquoA hierarchical approach forinfluential node ranking in complex social networksrdquo ExpertSystems with Applications vol 93 pp 200ndash211 2018 (inEnglish)

Complexity 13

evaluate the node importance +e core idea of TOPSIS-REis to construct a positive ideal object and a negative idealobject from the original data +e positive ideal object iscalculated based on the max value of the indicators and thenegative ideal object is calculated based on min value of the

indicators All methods are implemented by using thenetwork analysis software Cytoscape together with Javaprogramming language

In the experiments all nodes are ranked based on the nodeimportance+en the nodes are removed one by one from the

Table 1 Ranking result of BUPT campus network

Method Ranking resultsEBSAM 8 31 11 3 18 22 7 20 21 4 9 10 14 24 26 28TOPSIS-RE 8 18 31 3 11 7 21 20 9 10 4 14 24 26 28 22Random 19 12 23 25 30 32 13 34 16 5 27 21 9 20 6 10

6

18

9

19

20

10

12

7

282624

2725 2923

30

33

37

22

21

32

3435

36

13

16

15

14

17

1

4

2

5

(a)

6

109

19

12 282624

2725 2923

30

33

37

36

21

32

3435

13

16

15

14

17

1

4

2

5

(b)

6

109

19

121

5

2

282624

2725 2923

30

33

37

36

21

32

3435

13

16

15

14

17

(c)

65

2

19

121

2923

30

2725

33

37

36

21

32

3435

13

16

15

17

(d)

Figure 4 +e removing process with EBSAM on BUPTcampus network (a) 4 nodes have been removed (b) 8 nodes have been removed(c) 12 nodes have been removed (d) 16 nodes have been removed

02468

101214161820

1

NCC

3 5 7 9 11 13 15 17e number of nodes to be attacked

19 21 23 25 27 29 31 33 35 37

TOPSIS-REEBSAMRandom

Figure 5 Experiment results on BUPT campus network

Complexity 7

networks according to the ranking results +e Number ofConnected Components (NCC) is employed to evaluate theeffectiveness of the methods A connected component of anundirected network is a subgraph in which any two nodes areconnected to each other by edges After we remove one ormore nodes in a network the network will be divided intoseveral disconnected subgraphs Any node inside a subgraph isreachable from other nodes in the same subgraph +ere is nopath between two nodes belonging to different a subgraphNCC is the number of these disconnected subgraphs NCCreflects the connectivity of a network +e robustness of anetwork could be measured by calculating the size of thelargest connected component after removing a fraction of thenodes [32ndash34] +e number of connected component in anetwork could reflect its connectivity A larger value of NCCreflects that the network is divided into more disconnected

subgraphs which indicates the node you remove is moreimportant respect to network connectivity +erefore a largervalue of NCC indicates a better performance A node isconsidered to bemore important ifmore number of connectedcomponent increases after it has been removed

52 Experimental Results

521 Experiment Results on BUPT Campus Network+e topological structure of BUPT campus network is il-lustrated in Figure 2 +e number in each node is just anidentity of the node It does not have any meaning except toidentify different nodes +e node can be identified by itsnumber in the graph

+e node importance rank results of EBSAM TOPSIS-RE and a random selection algorithm (Random) are

Yun Cheng

Jin ChengLin Fen

Chang Zhi

Jin Zhong

Tai Yuan Yang Quan

Qi Zhou

Su Zhou

Da Tong

Figure 6 Shanxi water network

8 Complexity

0

5

10

15

20

25

30

35

1 3 5 7

NCC

9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43 45 47 49 51 53 55 57 59 61 63 65 67 69 71 73 75 77 79 81

TOPSIS-REEBSAMRandom

The number of nodes to be attacked

Figure 8 NCC of Shanxi water network

75

71

48

7473

81

68

8078

77

76

82

79

72

5655

70

69

57

54

51

5253

5049

44

33

4332

46

45

59

61 65

63

67

47

60

6664

62

58

16

35

27

14

28

34

221513

24 26

25

31

2021

10

12

2

9

1

38

36

29

39

1719

23

18

6

78

11

5

4

3

42

41

37

40

30

Figure 7 Topological structure of Shanxi water network

Complexity 9

illustrated in Table 1 We only list the top 16 nodes in therank results because the rank results of the rest nodes are thesame in EBSAM and TOPSIS-RE

According to the rank result we remove the nodes oneby one from the network until all nodes have been re-moved from the network We calculate the number ofconnected components in the network after removing anode +e removing process of EBSAM is shown in Fig-ure 4 We list out the topological structure of the networkafter removing every four nodes +e NCC of EBSAM andTOPSIS-RE and Random methods are shown in Figure 5As we can see the number of connected components ofRandom method is much less than the other two methods+erefore Random is less effective in destroying thenetwork by attacking the important nodes We can also seethat the number of connected components of EBSAM ismore than TOPSIS-RE Hence the connectivity of thenetwork is worse with EBSAM Attacking the network

based on the ranking result of EBSAM is more effectivethan TOPSIS-RE +at is because we obtain the weight ofthe four indicators objectively and adaptively other thanassign a fixed value subjectively

522 Experimental Results on Shanxi Water NetworkAs shown in Figure 6 Shanxi Water Network plays a vitalrole in the normal production and living activities+e greenline in Figure 6 denotes the water supply network +eShanxi water network provides guarantee for water demandof north China and its topological structure is shown inFigure 7 As shown in Figure 7 Shanxi Water Network iscomposed of 82 nodes +e experimental result is shown inFigure 8 As we can see the connectivity of the network hasbeen destroyed after the top 50 nodes have been attackedHowever NCC of our method is larger than the other twocompared methods +erefore the performance of EBSAMis better than other compared methods

Da Tong

Su Zhou

Qi Zhou

Tai Yuan Yang Quan

Jin Zhong

Lin Fen

Chang Zhi

Jin Cheng

Yun Cheng

The first 1000kmThe second 1000kmThe third 1000kmExpressway under construction and to be built

Figure 9 Shanxi railway network

10 Complexity

523 Experimental Results on Shanxi Railway NetworkFinally we conduct experiment on Shanxi Railway Net-work As shown in Figure 9 Shanxi Railway Network is apart of the transportation network in Shanxi It providesgreat convenience for peoplersquos outgoing and commodities

trading +e topological structure of Shanxi Water Net-work is shown in Figure 10 +e experimental result isshown in Figure 11 +e network is coming to break downafter the top 60 nodes have been attacked +e NCC ofShanxi Water Network obtains the largest ascent with our

4

31811

1

2

13

1219

2123

22

20

9

24

31

6

25

10

87

5

26

14

28

16 17

15

29

93

85

90

96

91

92

84

94

9597

98

33

27

30

3435

43

5150

3236

45

44

58

4138

40

3937

42

81

6465 66

7982

80 78

76

77

83

53

52

59

54

60

55

61

4948

46

47

86

62

87

63

57

89

56

886968

71

73 72

70

7475

67

Figure 10 Topological structure of Shanxi railway network

0

5

10

15

20

25

30

35

40

1 5 9 13 17

NCC

21 25 29 33 37 41 45 49 53 57 61 65 69 73 77 81 85 89 93 97

TOPSIS-REEBSAMRandom

The number of nodes to be attacked

Figure 11 NCC of Shanxi railway network

Complexity 11

method +erefore EBSAM performs better than othermethods

6 Conclusions and Future Work

In this paper we proposed an entropy theory-based self-adaptive node importance evaluation method for complexnetworks Firstly we select four centrality measures whichcan reflect different characteristics of the node as nodeimportance evaluation indicators +en we combine themtogether with appropriate weights calculated by an entropytheory-based algorithm +e algorithm shows a strongadaptability and thus allows be widely implemented indifferent kinds of networks In the traditional method theweights are selected based on the subjective experience of theresearchers instead of enough scientific basis which wouldlead to inaccurate evaluation results +e proposed methodis better because it utilizes entropy theory to calculate theweight of each indicator A smaller value of entropy indicatesthat the corresponding attribution contains less useful in-formation In a multiattribute decision-making problem weneed to assign a larger weight to attribute with more usefulinformation rather than the attribute with greater uncer-tainty So with this algorithm we can better assign properweight to different attributions +e experimental results onthree types of real-world complex networks show that ourmethod performs better with compared methods Our on-going research will focus on investigating the effectiveness ofour method in more complex environments

Data Availability

+e data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

+e authors declare that they have no conflicts of interest

Acknowledgments

+is study was supported by National Key RampD Program ofChina (Funding No 2018YFB1402800) and the NaturalScience Foundation of China (No 61571066)

References

[1] SWang X Zhang Y Zhang LWang J Yang andWWangldquoA survey on mobile edge networks convergence of com-puting caching and communicationsrdquo Ieee Access vol 5pp 6757ndash6779 2017 (in English)

[2] X Ren and L Lu ldquoReview of ranking nodes in complexnetworksrdquo Chinese Science Bulletin vol 59 no 13pp 1175ndash1197 2014

[3] K L S Samanta and M Pal ldquoStudy on centrality measures insocial networks a surveyrdquo Social Network Analysis Miningvol 8 no 1 p 13 2018

[4] J Dong F Ye W Chen and J Wu ldquoIdentifying influentialnodes in complex networks via semi-local centralityrdquo inProceedings of the 2018 IEEE International Symposium on

Circuits and Systems (Iscas) (in English) Florence Italy May2018

[5] K Angelou M Maragakis and P Argyrakis ldquoA structuralanalysis of the patent citation network by the k-shell de-composition methodrdquo Physica A-Statistical Mechanics and ItsApplications vol 521 pp 476ndash483 2019 (in English)

[6] G Li M Li J Wang Y Li and Y Pan ldquoUnited neighborhoodcloseness centrality and orthology for predicting essentialproteinsrdquo IEEEACM Transactions on Computational Biologyand Bioinformatics 2018

[7] M Riondato and E Upfal ldquoABRA approximating be-tweenness centrality in static and dynamic graphs withrademacher averagesrdquo ACM Transactions on KnowledgeDiscovery from Data vol 12 no 5 pp 1ndash38 2018 (inEnglish)

[8] N Gupta A Singh and H Cherifi ldquoCentrality measures fornetworks with community structurerdquo Physica A StatisticalMechanics and Its Applications vol 452 pp 46ndash59 2016 (inEnglish)

[9] D Chen H Gao L Lu and T Zhou ldquoIdentifying influentialnodes in large-scale directed networks the role of clusteringrdquoPLoS One vol 8 Article ID e77455 no 10 (in English) 2013

[10] K Tu P Cui X Wang P Yu and W Zhu ldquoDeep recursivenetwork embedding with regular equivalencerdquo in Proceedingsof the Kddrsquo18 24th Acm Sigkdd International Conference onKnowledge Discovery amp Data Mining London UKpp 2357ndash2366 August 2018 (in English)

[11] R Poulin M-C Boily and B R Masse ldquoDynamical systemsto define centrality in social networksrdquo Social Networksvol 22 no 3 pp 187ndash220 2000 (in English)

[12] P Hu and T Mei ldquoRanking influential nodes in complexnetworks with structural holesrdquo Physica A Statistical Me-chanics and Its Applications vol 490 pp 624ndash631 15 2018 (inEnglish)

[13] L Lu Y Zhang C Yeung and T Zhou ldquoLeaders in socialnetworks the delicious caserdquo Plos One vol 6 no 6 (inEnglish) 2011

[14] J Liu Q Xiong W Shi X Shi and K Wang ldquoEvaluating theimportance of nodes in complex networksrdquo Physica A Sta-tistical Mechanics and Its Applications vol 452 pp 209ndash2192016 (in English)

[15] F Hu Y Liu and J Jin ldquoMulti-index evaluation algorithmbased on locally linear embedding for the node importance incomplex networksrdquo in Proceedings of ampirteenth InternationalSymposium on Distributed Computing and Applications toBusiness Engineering and Science (Dcabes 2014) pp 138ndash142(in English) Xian Ning China November 2014

[16] S T Roweis and L K Saul ldquoNonlinear dimensionality re-duction by locally linear embeddingrdquo Science vol 290no 5500 pp 2323ndash2326 2000 (in English)

[17] L Xu J Liu Y Liu Y Liu J Gou and B Masoud ldquoNodeimportance classified comprehensive assessmentrdquo Proceed-ings of the Csee vol 34 no 10 pp 1609ndash1617 2014

[18] X Zhang and X Wang ldquoEvaluation formula for communi-cation network node importancerdquo Journal of NortheasternUniversity vol 35 no 5 pp 663ndash666 2014

[19] H Ping W Fan and S Mei ldquoIdentifying node importance incomplex networksrdquo Physica A Statistical Mechanics and ItsApplications vol 429 pp 169ndash176 2015

[20] Y Yu ldquoNode importance measurement based on the degreeand closeness centralityrdquo Journal of Information and Com-putational Science vol 12 no 3 pp 1281ndash1291 2015

12 Complexity

[21] Q Hu Y Yin P Ma Y Gao Y Zhang and C Xing ldquoA newapproach to identify influential spreaders in complex net-worksrdquoActa Physica Sinica vol 62 no 14 (in Chinese) 2013

[22] Y Zhang Y Bao S Zhao J Chen and J Tang ldquoIdentifyingnode importance by combining betweenness centrality andKatz centralityrdquo in Proceedings of the 2015 InternationalConference on Cloud Computing and Big Data (Ccbd)pp 354ndash357 (in English) Shanghai China November 2015

[23] Y Yang and G Xie ldquoEfficient identification of node im-portance in social networksrdquo Information Processing ampManagement vol 52 no 5 pp 911ndash922 2016 (in English)

[24] Z Liu C Jiang J Wang and H Yu ldquo+e node importance inactual complex networks based on a multi-attribute rankingmethodrdquo Knowledge-Based Systems vol 84 pp 56ndash66 2015(in English)

[25] C E Shannon ldquoA mathematical theory of communicationrdquoBell System Technical Journal vol 27 no 4 pp 623ndash656 1948

[26] Q Zhang M Li and Y Deng ldquoMeasure the structure sim-ilarity of nodes in complex networks based on relative en-tropyrdquo Physica A Statistical Mechanics and Its Applicationsvol 491 pp 749ndash763 2018 (in English)

[27] Y He H Guo M Jin and P Ren ldquoA linguistic entropyweight method and its application in linguistic multi-attributegroup decision makingrdquo Nonlinear Dynamics vol 84 no 1pp 399ndash404 2016 (in English)

[28] X Liang and C Wei ldquoAn Atanassovrsquos intuitionistic fuzzymulti-attribute group decision making method based onentropy and similarity measurerdquo International Journal ofMachine Learning and Cybernetics vol 5 no 3 pp 435ndash4442014 (in English)

[29] T-Y Chen and C-H Li ldquoObjective weights with intuition-istic fuzzy entropy measures and computational experimentanalysisrdquo Applied Soft Computing vol 11 no 8 pp 5411ndash5423 2011 (in English)

[30] S-P Wan Q-Y Wang and J-Y Dong ldquo+e extendedVIKOR method for multi-attribute group decision makingwith triangular intuitionistic fuzzy numbersrdquo Knowledge-Based Systems vol 52 pp 65ndash77 2013 (in English)

[31] H Nguyen ldquoA new knowledge-based measure for intui-tionistic fuzzy sets and its application in multiple attributegroup decision makingrdquo Expert Systems with Applicationsvol 42 no 22 pp 8766ndash8774 2015 (in English)

[32] S Dereich and P Morters ldquoRandom networks with sublinearpreferential attachment the giant componentrdquoampe Annals ofProbability vol 41 no 1 pp 329ndash384 2013 (in English)

[33] C M Schneider A A Moreira J S Andrade S Havlin andH J Herrmann ldquoMitigation of malicious attacks on net-worksrdquo Proceedings of the National Academy of Sciences of theUnited States of America vol 108 no 10 pp 3838ndash3841 2011(in English)

[34] A Zareie and A Sheikhahmadi ldquoA hierarchical approach forinfluential node ranking in complex social networksrdquo ExpertSystems with Applications vol 93 pp 200ndash211 2018 (inEnglish)

Complexity 13

networks according to the ranking results +e Number ofConnected Components (NCC) is employed to evaluate theeffectiveness of the methods A connected component of anundirected network is a subgraph in which any two nodes areconnected to each other by edges After we remove one ormore nodes in a network the network will be divided intoseveral disconnected subgraphs Any node inside a subgraph isreachable from other nodes in the same subgraph +ere is nopath between two nodes belonging to different a subgraphNCC is the number of these disconnected subgraphs NCCreflects the connectivity of a network +e robustness of anetwork could be measured by calculating the size of thelargest connected component after removing a fraction of thenodes [32ndash34] +e number of connected component in anetwork could reflect its connectivity A larger value of NCCreflects that the network is divided into more disconnected

subgraphs which indicates the node you remove is moreimportant respect to network connectivity +erefore a largervalue of NCC indicates a better performance A node isconsidered to bemore important ifmore number of connectedcomponent increases after it has been removed

52 Experimental Results

521 Experiment Results on BUPT Campus Network+e topological structure of BUPT campus network is il-lustrated in Figure 2 +e number in each node is just anidentity of the node It does not have any meaning except toidentify different nodes +e node can be identified by itsnumber in the graph

+e node importance rank results of EBSAM TOPSIS-RE and a random selection algorithm (Random) are

Yun Cheng

Jin ChengLin Fen

Chang Zhi

Jin Zhong

Tai Yuan Yang Quan

Qi Zhou

Su Zhou

Da Tong

Figure 6 Shanxi water network

8 Complexity

0

5

10

15

20

25

30

35

1 3 5 7

NCC

9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43 45 47 49 51 53 55 57 59 61 63 65 67 69 71 73 75 77 79 81

TOPSIS-REEBSAMRandom

The number of nodes to be attacked

Figure 8 NCC of Shanxi water network

75

71

48

7473

81

68

8078

77

76

82

79

72

5655

70

69

57

54

51

5253

5049

44

33

4332

46

45

59

61 65

63

67

47

60

6664

62

58

16

35

27

14

28

34

221513

24 26

25

31

2021

10

12

2

9

1

38

36

29

39

1719

23

18

6

78

11

5

4

3

42

41

37

40

30

Figure 7 Topological structure of Shanxi water network

Complexity 9

illustrated in Table 1 We only list the top 16 nodes in therank results because the rank results of the rest nodes are thesame in EBSAM and TOPSIS-RE

According to the rank result we remove the nodes oneby one from the network until all nodes have been re-moved from the network We calculate the number ofconnected components in the network after removing anode +e removing process of EBSAM is shown in Fig-ure 4 We list out the topological structure of the networkafter removing every four nodes +e NCC of EBSAM andTOPSIS-RE and Random methods are shown in Figure 5As we can see the number of connected components ofRandom method is much less than the other two methods+erefore Random is less effective in destroying thenetwork by attacking the important nodes We can also seethat the number of connected components of EBSAM ismore than TOPSIS-RE Hence the connectivity of thenetwork is worse with EBSAM Attacking the network

based on the ranking result of EBSAM is more effectivethan TOPSIS-RE +at is because we obtain the weight ofthe four indicators objectively and adaptively other thanassign a fixed value subjectively

522 Experimental Results on Shanxi Water NetworkAs shown in Figure 6 Shanxi Water Network plays a vitalrole in the normal production and living activities+e greenline in Figure 6 denotes the water supply network +eShanxi water network provides guarantee for water demandof north China and its topological structure is shown inFigure 7 As shown in Figure 7 Shanxi Water Network iscomposed of 82 nodes +e experimental result is shown inFigure 8 As we can see the connectivity of the network hasbeen destroyed after the top 50 nodes have been attackedHowever NCC of our method is larger than the other twocompared methods +erefore the performance of EBSAMis better than other compared methods

Da Tong

Su Zhou

Qi Zhou

Tai Yuan Yang Quan

Jin Zhong

Lin Fen

Chang Zhi

Jin Cheng

Yun Cheng

The first 1000kmThe second 1000kmThe third 1000kmExpressway under construction and to be built

Figure 9 Shanxi railway network

10 Complexity

523 Experimental Results on Shanxi Railway NetworkFinally we conduct experiment on Shanxi Railway Net-work As shown in Figure 9 Shanxi Railway Network is apart of the transportation network in Shanxi It providesgreat convenience for peoplersquos outgoing and commodities

trading +e topological structure of Shanxi Water Net-work is shown in Figure 10 +e experimental result isshown in Figure 11 +e network is coming to break downafter the top 60 nodes have been attacked +e NCC ofShanxi Water Network obtains the largest ascent with our

4

31811

1

2

13

1219

2123

22

20

9

24

31

6

25

10

87

5

26

14

28

16 17

15

29

93

85

90

96

91

92

84

94

9597

98

33

27

30

3435

43

5150

3236

45

44

58

4138

40

3937

42

81

6465 66

7982

80 78

76

77

83

53

52

59

54

60

55

61

4948

46

47

86

62

87

63

57

89

56

886968

71

73 72

70

7475

67

Figure 10 Topological structure of Shanxi railway network

0

5

10

15

20

25

30

35

40

1 5 9 13 17

NCC

21 25 29 33 37 41 45 49 53 57 61 65 69 73 77 81 85 89 93 97

TOPSIS-REEBSAMRandom

The number of nodes to be attacked

Figure 11 NCC of Shanxi railway network

Complexity 11

method +erefore EBSAM performs better than othermethods

6 Conclusions and Future Work

In this paper we proposed an entropy theory-based self-adaptive node importance evaluation method for complexnetworks Firstly we select four centrality measures whichcan reflect different characteristics of the node as nodeimportance evaluation indicators +en we combine themtogether with appropriate weights calculated by an entropytheory-based algorithm +e algorithm shows a strongadaptability and thus allows be widely implemented indifferent kinds of networks In the traditional method theweights are selected based on the subjective experience of theresearchers instead of enough scientific basis which wouldlead to inaccurate evaluation results +e proposed methodis better because it utilizes entropy theory to calculate theweight of each indicator A smaller value of entropy indicatesthat the corresponding attribution contains less useful in-formation In a multiattribute decision-making problem weneed to assign a larger weight to attribute with more usefulinformation rather than the attribute with greater uncer-tainty So with this algorithm we can better assign properweight to different attributions +e experimental results onthree types of real-world complex networks show that ourmethod performs better with compared methods Our on-going research will focus on investigating the effectiveness ofour method in more complex environments

Data Availability

+e data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

+e authors declare that they have no conflicts of interest

Acknowledgments

+is study was supported by National Key RampD Program ofChina (Funding No 2018YFB1402800) and the NaturalScience Foundation of China (No 61571066)

References

[1] SWang X Zhang Y Zhang LWang J Yang andWWangldquoA survey on mobile edge networks convergence of com-puting caching and communicationsrdquo Ieee Access vol 5pp 6757ndash6779 2017 (in English)

[2] X Ren and L Lu ldquoReview of ranking nodes in complexnetworksrdquo Chinese Science Bulletin vol 59 no 13pp 1175ndash1197 2014

[3] K L S Samanta and M Pal ldquoStudy on centrality measures insocial networks a surveyrdquo Social Network Analysis Miningvol 8 no 1 p 13 2018

[4] J Dong F Ye W Chen and J Wu ldquoIdentifying influentialnodes in complex networks via semi-local centralityrdquo inProceedings of the 2018 IEEE International Symposium on

Circuits and Systems (Iscas) (in English) Florence Italy May2018

[5] K Angelou M Maragakis and P Argyrakis ldquoA structuralanalysis of the patent citation network by the k-shell de-composition methodrdquo Physica A-Statistical Mechanics and ItsApplications vol 521 pp 476ndash483 2019 (in English)

[6] G Li M Li J Wang Y Li and Y Pan ldquoUnited neighborhoodcloseness centrality and orthology for predicting essentialproteinsrdquo IEEEACM Transactions on Computational Biologyand Bioinformatics 2018

[7] M Riondato and E Upfal ldquoABRA approximating be-tweenness centrality in static and dynamic graphs withrademacher averagesrdquo ACM Transactions on KnowledgeDiscovery from Data vol 12 no 5 pp 1ndash38 2018 (inEnglish)

[8] N Gupta A Singh and H Cherifi ldquoCentrality measures fornetworks with community structurerdquo Physica A StatisticalMechanics and Its Applications vol 452 pp 46ndash59 2016 (inEnglish)

[9] D Chen H Gao L Lu and T Zhou ldquoIdentifying influentialnodes in large-scale directed networks the role of clusteringrdquoPLoS One vol 8 Article ID e77455 no 10 (in English) 2013

[10] K Tu P Cui X Wang P Yu and W Zhu ldquoDeep recursivenetwork embedding with regular equivalencerdquo in Proceedingsof the Kddrsquo18 24th Acm Sigkdd International Conference onKnowledge Discovery amp Data Mining London UKpp 2357ndash2366 August 2018 (in English)

[11] R Poulin M-C Boily and B R Masse ldquoDynamical systemsto define centrality in social networksrdquo Social Networksvol 22 no 3 pp 187ndash220 2000 (in English)

[12] P Hu and T Mei ldquoRanking influential nodes in complexnetworks with structural holesrdquo Physica A Statistical Me-chanics and Its Applications vol 490 pp 624ndash631 15 2018 (inEnglish)

[13] L Lu Y Zhang C Yeung and T Zhou ldquoLeaders in socialnetworks the delicious caserdquo Plos One vol 6 no 6 (inEnglish) 2011

[14] J Liu Q Xiong W Shi X Shi and K Wang ldquoEvaluating theimportance of nodes in complex networksrdquo Physica A Sta-tistical Mechanics and Its Applications vol 452 pp 209ndash2192016 (in English)

[15] F Hu Y Liu and J Jin ldquoMulti-index evaluation algorithmbased on locally linear embedding for the node importance incomplex networksrdquo in Proceedings of ampirteenth InternationalSymposium on Distributed Computing and Applications toBusiness Engineering and Science (Dcabes 2014) pp 138ndash142(in English) Xian Ning China November 2014

[16] S T Roweis and L K Saul ldquoNonlinear dimensionality re-duction by locally linear embeddingrdquo Science vol 290no 5500 pp 2323ndash2326 2000 (in English)

[17] L Xu J Liu Y Liu Y Liu J Gou and B Masoud ldquoNodeimportance classified comprehensive assessmentrdquo Proceed-ings of the Csee vol 34 no 10 pp 1609ndash1617 2014

[18] X Zhang and X Wang ldquoEvaluation formula for communi-cation network node importancerdquo Journal of NortheasternUniversity vol 35 no 5 pp 663ndash666 2014

[19] H Ping W Fan and S Mei ldquoIdentifying node importance incomplex networksrdquo Physica A Statistical Mechanics and ItsApplications vol 429 pp 169ndash176 2015

[20] Y Yu ldquoNode importance measurement based on the degreeand closeness centralityrdquo Journal of Information and Com-putational Science vol 12 no 3 pp 1281ndash1291 2015

12 Complexity

[21] Q Hu Y Yin P Ma Y Gao Y Zhang and C Xing ldquoA newapproach to identify influential spreaders in complex net-worksrdquoActa Physica Sinica vol 62 no 14 (in Chinese) 2013

[22] Y Zhang Y Bao S Zhao J Chen and J Tang ldquoIdentifyingnode importance by combining betweenness centrality andKatz centralityrdquo in Proceedings of the 2015 InternationalConference on Cloud Computing and Big Data (Ccbd)pp 354ndash357 (in English) Shanghai China November 2015

[23] Y Yang and G Xie ldquoEfficient identification of node im-portance in social networksrdquo Information Processing ampManagement vol 52 no 5 pp 911ndash922 2016 (in English)

[24] Z Liu C Jiang J Wang and H Yu ldquo+e node importance inactual complex networks based on a multi-attribute rankingmethodrdquo Knowledge-Based Systems vol 84 pp 56ndash66 2015(in English)

[25] C E Shannon ldquoA mathematical theory of communicationrdquoBell System Technical Journal vol 27 no 4 pp 623ndash656 1948

[26] Q Zhang M Li and Y Deng ldquoMeasure the structure sim-ilarity of nodes in complex networks based on relative en-tropyrdquo Physica A Statistical Mechanics and Its Applicationsvol 491 pp 749ndash763 2018 (in English)

[27] Y He H Guo M Jin and P Ren ldquoA linguistic entropyweight method and its application in linguistic multi-attributegroup decision makingrdquo Nonlinear Dynamics vol 84 no 1pp 399ndash404 2016 (in English)

[28] X Liang and C Wei ldquoAn Atanassovrsquos intuitionistic fuzzymulti-attribute group decision making method based onentropy and similarity measurerdquo International Journal ofMachine Learning and Cybernetics vol 5 no 3 pp 435ndash4442014 (in English)

[29] T-Y Chen and C-H Li ldquoObjective weights with intuition-istic fuzzy entropy measures and computational experimentanalysisrdquo Applied Soft Computing vol 11 no 8 pp 5411ndash5423 2011 (in English)

[30] S-P Wan Q-Y Wang and J-Y Dong ldquo+e extendedVIKOR method for multi-attribute group decision makingwith triangular intuitionistic fuzzy numbersrdquo Knowledge-Based Systems vol 52 pp 65ndash77 2013 (in English)

[31] H Nguyen ldquoA new knowledge-based measure for intui-tionistic fuzzy sets and its application in multiple attributegroup decision makingrdquo Expert Systems with Applicationsvol 42 no 22 pp 8766ndash8774 2015 (in English)

[32] S Dereich and P Morters ldquoRandom networks with sublinearpreferential attachment the giant componentrdquoampe Annals ofProbability vol 41 no 1 pp 329ndash384 2013 (in English)

[33] C M Schneider A A Moreira J S Andrade S Havlin andH J Herrmann ldquoMitigation of malicious attacks on net-worksrdquo Proceedings of the National Academy of Sciences of theUnited States of America vol 108 no 10 pp 3838ndash3841 2011(in English)

[34] A Zareie and A Sheikhahmadi ldquoA hierarchical approach forinfluential node ranking in complex social networksrdquo ExpertSystems with Applications vol 93 pp 200ndash211 2018 (inEnglish)

Complexity 13

0

5

10

15

20

25

30

35

1 3 5 7

NCC

9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43 45 47 49 51 53 55 57 59 61 63 65 67 69 71 73 75 77 79 81

TOPSIS-REEBSAMRandom

The number of nodes to be attacked

Figure 8 NCC of Shanxi water network

75

71

48

7473

81

68

8078

77

76

82

79

72

5655

70

69

57

54

51

5253

5049

44

33

4332

46

45

59

61 65

63

67

47

60

6664

62

58

16

35

27

14

28

34

221513

24 26

25

31

2021

10

12

2

9

1

38

36

29

39

1719

23

18

6

78

11

5

4

3

42

41

37

40

30

Figure 7 Topological structure of Shanxi water network

Complexity 9

illustrated in Table 1 We only list the top 16 nodes in therank results because the rank results of the rest nodes are thesame in EBSAM and TOPSIS-RE

According to the rank result we remove the nodes oneby one from the network until all nodes have been re-moved from the network We calculate the number ofconnected components in the network after removing anode +e removing process of EBSAM is shown in Fig-ure 4 We list out the topological structure of the networkafter removing every four nodes +e NCC of EBSAM andTOPSIS-RE and Random methods are shown in Figure 5As we can see the number of connected components ofRandom method is much less than the other two methods+erefore Random is less effective in destroying thenetwork by attacking the important nodes We can also seethat the number of connected components of EBSAM ismore than TOPSIS-RE Hence the connectivity of thenetwork is worse with EBSAM Attacking the network

based on the ranking result of EBSAM is more effectivethan TOPSIS-RE +at is because we obtain the weight ofthe four indicators objectively and adaptively other thanassign a fixed value subjectively

522 Experimental Results on Shanxi Water NetworkAs shown in Figure 6 Shanxi Water Network plays a vitalrole in the normal production and living activities+e greenline in Figure 6 denotes the water supply network +eShanxi water network provides guarantee for water demandof north China and its topological structure is shown inFigure 7 As shown in Figure 7 Shanxi Water Network iscomposed of 82 nodes +e experimental result is shown inFigure 8 As we can see the connectivity of the network hasbeen destroyed after the top 50 nodes have been attackedHowever NCC of our method is larger than the other twocompared methods +erefore the performance of EBSAMis better than other compared methods

Da Tong

Su Zhou

Qi Zhou

Tai Yuan Yang Quan

Jin Zhong

Lin Fen

Chang Zhi

Jin Cheng

Yun Cheng

The first 1000kmThe second 1000kmThe third 1000kmExpressway under construction and to be built

Figure 9 Shanxi railway network

10 Complexity

523 Experimental Results on Shanxi Railway NetworkFinally we conduct experiment on Shanxi Railway Net-work As shown in Figure 9 Shanxi Railway Network is apart of the transportation network in Shanxi It providesgreat convenience for peoplersquos outgoing and commodities

trading +e topological structure of Shanxi Water Net-work is shown in Figure 10 +e experimental result isshown in Figure 11 +e network is coming to break downafter the top 60 nodes have been attacked +e NCC ofShanxi Water Network obtains the largest ascent with our

4

31811

1

2

13

1219

2123

22

20

9

24

31

6

25

10

87

5

26

14

28

16 17

15

29

93

85

90

96

91

92

84

94

9597

98

33

27

30

3435

43

5150

3236

45

44

58

4138

40

3937

42

81

6465 66

7982

80 78

76

77

83

53

52

59

54

60

55

61

4948

46

47

86

62

87

63

57

89

56

886968

71

73 72

70

7475

67

Figure 10 Topological structure of Shanxi railway network

0

5

10

15

20

25

30

35

40

1 5 9 13 17

NCC

21 25 29 33 37 41 45 49 53 57 61 65 69 73 77 81 85 89 93 97

TOPSIS-REEBSAMRandom

The number of nodes to be attacked

Figure 11 NCC of Shanxi railway network

Complexity 11

method +erefore EBSAM performs better than othermethods

6 Conclusions and Future Work

In this paper we proposed an entropy theory-based self-adaptive node importance evaluation method for complexnetworks Firstly we select four centrality measures whichcan reflect different characteristics of the node as nodeimportance evaluation indicators +en we combine themtogether with appropriate weights calculated by an entropytheory-based algorithm +e algorithm shows a strongadaptability and thus allows be widely implemented indifferent kinds of networks In the traditional method theweights are selected based on the subjective experience of theresearchers instead of enough scientific basis which wouldlead to inaccurate evaluation results +e proposed methodis better because it utilizes entropy theory to calculate theweight of each indicator A smaller value of entropy indicatesthat the corresponding attribution contains less useful in-formation In a multiattribute decision-making problem weneed to assign a larger weight to attribute with more usefulinformation rather than the attribute with greater uncer-tainty So with this algorithm we can better assign properweight to different attributions +e experimental results onthree types of real-world complex networks show that ourmethod performs better with compared methods Our on-going research will focus on investigating the effectiveness ofour method in more complex environments

Data Availability

+e data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

+e authors declare that they have no conflicts of interest

Acknowledgments

+is study was supported by National Key RampD Program ofChina (Funding No 2018YFB1402800) and the NaturalScience Foundation of China (No 61571066)

References

[1] SWang X Zhang Y Zhang LWang J Yang andWWangldquoA survey on mobile edge networks convergence of com-puting caching and communicationsrdquo Ieee Access vol 5pp 6757ndash6779 2017 (in English)

[2] X Ren and L Lu ldquoReview of ranking nodes in complexnetworksrdquo Chinese Science Bulletin vol 59 no 13pp 1175ndash1197 2014

[3] K L S Samanta and M Pal ldquoStudy on centrality measures insocial networks a surveyrdquo Social Network Analysis Miningvol 8 no 1 p 13 2018

[4] J Dong F Ye W Chen and J Wu ldquoIdentifying influentialnodes in complex networks via semi-local centralityrdquo inProceedings of the 2018 IEEE International Symposium on

Circuits and Systems (Iscas) (in English) Florence Italy May2018

[5] K Angelou M Maragakis and P Argyrakis ldquoA structuralanalysis of the patent citation network by the k-shell de-composition methodrdquo Physica A-Statistical Mechanics and ItsApplications vol 521 pp 476ndash483 2019 (in English)

[6] G Li M Li J Wang Y Li and Y Pan ldquoUnited neighborhoodcloseness centrality and orthology for predicting essentialproteinsrdquo IEEEACM Transactions on Computational Biologyand Bioinformatics 2018

[7] M Riondato and E Upfal ldquoABRA approximating be-tweenness centrality in static and dynamic graphs withrademacher averagesrdquo ACM Transactions on KnowledgeDiscovery from Data vol 12 no 5 pp 1ndash38 2018 (inEnglish)

[8] N Gupta A Singh and H Cherifi ldquoCentrality measures fornetworks with community structurerdquo Physica A StatisticalMechanics and Its Applications vol 452 pp 46ndash59 2016 (inEnglish)

[9] D Chen H Gao L Lu and T Zhou ldquoIdentifying influentialnodes in large-scale directed networks the role of clusteringrdquoPLoS One vol 8 Article ID e77455 no 10 (in English) 2013

[10] K Tu P Cui X Wang P Yu and W Zhu ldquoDeep recursivenetwork embedding with regular equivalencerdquo in Proceedingsof the Kddrsquo18 24th Acm Sigkdd International Conference onKnowledge Discovery amp Data Mining London UKpp 2357ndash2366 August 2018 (in English)

[11] R Poulin M-C Boily and B R Masse ldquoDynamical systemsto define centrality in social networksrdquo Social Networksvol 22 no 3 pp 187ndash220 2000 (in English)

[12] P Hu and T Mei ldquoRanking influential nodes in complexnetworks with structural holesrdquo Physica A Statistical Me-chanics and Its Applications vol 490 pp 624ndash631 15 2018 (inEnglish)

[13] L Lu Y Zhang C Yeung and T Zhou ldquoLeaders in socialnetworks the delicious caserdquo Plos One vol 6 no 6 (inEnglish) 2011

[14] J Liu Q Xiong W Shi X Shi and K Wang ldquoEvaluating theimportance of nodes in complex networksrdquo Physica A Sta-tistical Mechanics and Its Applications vol 452 pp 209ndash2192016 (in English)

[15] F Hu Y Liu and J Jin ldquoMulti-index evaluation algorithmbased on locally linear embedding for the node importance incomplex networksrdquo in Proceedings of ampirteenth InternationalSymposium on Distributed Computing and Applications toBusiness Engineering and Science (Dcabes 2014) pp 138ndash142(in English) Xian Ning China November 2014

[16] S T Roweis and L K Saul ldquoNonlinear dimensionality re-duction by locally linear embeddingrdquo Science vol 290no 5500 pp 2323ndash2326 2000 (in English)

[17] L Xu J Liu Y Liu Y Liu J Gou and B Masoud ldquoNodeimportance classified comprehensive assessmentrdquo Proceed-ings of the Csee vol 34 no 10 pp 1609ndash1617 2014

[18] X Zhang and X Wang ldquoEvaluation formula for communi-cation network node importancerdquo Journal of NortheasternUniversity vol 35 no 5 pp 663ndash666 2014

[19] H Ping W Fan and S Mei ldquoIdentifying node importance incomplex networksrdquo Physica A Statistical Mechanics and ItsApplications vol 429 pp 169ndash176 2015

[20] Y Yu ldquoNode importance measurement based on the degreeand closeness centralityrdquo Journal of Information and Com-putational Science vol 12 no 3 pp 1281ndash1291 2015

12 Complexity

[21] Q Hu Y Yin P Ma Y Gao Y Zhang and C Xing ldquoA newapproach to identify influential spreaders in complex net-worksrdquoActa Physica Sinica vol 62 no 14 (in Chinese) 2013

[22] Y Zhang Y Bao S Zhao J Chen and J Tang ldquoIdentifyingnode importance by combining betweenness centrality andKatz centralityrdquo in Proceedings of the 2015 InternationalConference on Cloud Computing and Big Data (Ccbd)pp 354ndash357 (in English) Shanghai China November 2015

[23] Y Yang and G Xie ldquoEfficient identification of node im-portance in social networksrdquo Information Processing ampManagement vol 52 no 5 pp 911ndash922 2016 (in English)

[24] Z Liu C Jiang J Wang and H Yu ldquo+e node importance inactual complex networks based on a multi-attribute rankingmethodrdquo Knowledge-Based Systems vol 84 pp 56ndash66 2015(in English)

[25] C E Shannon ldquoA mathematical theory of communicationrdquoBell System Technical Journal vol 27 no 4 pp 623ndash656 1948

[26] Q Zhang M Li and Y Deng ldquoMeasure the structure sim-ilarity of nodes in complex networks based on relative en-tropyrdquo Physica A Statistical Mechanics and Its Applicationsvol 491 pp 749ndash763 2018 (in English)

[27] Y He H Guo M Jin and P Ren ldquoA linguistic entropyweight method and its application in linguistic multi-attributegroup decision makingrdquo Nonlinear Dynamics vol 84 no 1pp 399ndash404 2016 (in English)

[28] X Liang and C Wei ldquoAn Atanassovrsquos intuitionistic fuzzymulti-attribute group decision making method based onentropy and similarity measurerdquo International Journal ofMachine Learning and Cybernetics vol 5 no 3 pp 435ndash4442014 (in English)

[29] T-Y Chen and C-H Li ldquoObjective weights with intuition-istic fuzzy entropy measures and computational experimentanalysisrdquo Applied Soft Computing vol 11 no 8 pp 5411ndash5423 2011 (in English)

[30] S-P Wan Q-Y Wang and J-Y Dong ldquo+e extendedVIKOR method for multi-attribute group decision makingwith triangular intuitionistic fuzzy numbersrdquo Knowledge-Based Systems vol 52 pp 65ndash77 2013 (in English)

[31] H Nguyen ldquoA new knowledge-based measure for intui-tionistic fuzzy sets and its application in multiple attributegroup decision makingrdquo Expert Systems with Applicationsvol 42 no 22 pp 8766ndash8774 2015 (in English)

[32] S Dereich and P Morters ldquoRandom networks with sublinearpreferential attachment the giant componentrdquoampe Annals ofProbability vol 41 no 1 pp 329ndash384 2013 (in English)

[33] C M Schneider A A Moreira J S Andrade S Havlin andH J Herrmann ldquoMitigation of malicious attacks on net-worksrdquo Proceedings of the National Academy of Sciences of theUnited States of America vol 108 no 10 pp 3838ndash3841 2011(in English)

[34] A Zareie and A Sheikhahmadi ldquoA hierarchical approach forinfluential node ranking in complex social networksrdquo ExpertSystems with Applications vol 93 pp 200ndash211 2018 (inEnglish)

Complexity 13

illustrated in Table 1 We only list the top 16 nodes in therank results because the rank results of the rest nodes are thesame in EBSAM and TOPSIS-RE

According to the rank result we remove the nodes oneby one from the network until all nodes have been re-moved from the network We calculate the number ofconnected components in the network after removing anode +e removing process of EBSAM is shown in Fig-ure 4 We list out the topological structure of the networkafter removing every four nodes +e NCC of EBSAM andTOPSIS-RE and Random methods are shown in Figure 5As we can see the number of connected components ofRandom method is much less than the other two methods+erefore Random is less effective in destroying thenetwork by attacking the important nodes We can also seethat the number of connected components of EBSAM ismore than TOPSIS-RE Hence the connectivity of thenetwork is worse with EBSAM Attacking the network

based on the ranking result of EBSAM is more effectivethan TOPSIS-RE +at is because we obtain the weight ofthe four indicators objectively and adaptively other thanassign a fixed value subjectively

522 Experimental Results on Shanxi Water NetworkAs shown in Figure 6 Shanxi Water Network plays a vitalrole in the normal production and living activities+e greenline in Figure 6 denotes the water supply network +eShanxi water network provides guarantee for water demandof north China and its topological structure is shown inFigure 7 As shown in Figure 7 Shanxi Water Network iscomposed of 82 nodes +e experimental result is shown inFigure 8 As we can see the connectivity of the network hasbeen destroyed after the top 50 nodes have been attackedHowever NCC of our method is larger than the other twocompared methods +erefore the performance of EBSAMis better than other compared methods

Da Tong

Su Zhou

Qi Zhou

Tai Yuan Yang Quan

Jin Zhong

Lin Fen

Chang Zhi

Jin Cheng

Yun Cheng

The first 1000kmThe second 1000kmThe third 1000kmExpressway under construction and to be built

Figure 9 Shanxi railway network

10 Complexity

523 Experimental Results on Shanxi Railway NetworkFinally we conduct experiment on Shanxi Railway Net-work As shown in Figure 9 Shanxi Railway Network is apart of the transportation network in Shanxi It providesgreat convenience for peoplersquos outgoing and commodities

trading +e topological structure of Shanxi Water Net-work is shown in Figure 10 +e experimental result isshown in Figure 11 +e network is coming to break downafter the top 60 nodes have been attacked +e NCC ofShanxi Water Network obtains the largest ascent with our

4

31811

1

2

13

1219

2123

22

20

9

24

31

6

25

10

87

5

26

14

28

16 17

15

29

93

85

90

96

91

92

84

94

9597

98

33

27

30

3435

43

5150

3236

45

44

58

4138

40

3937

42

81

6465 66

7982

80 78

76

77

83

53

52

59

54

60

55

61

4948

46

47

86

62

87

63

57

89

56

886968

71

73 72

70

7475

67

Figure 10 Topological structure of Shanxi railway network

0

5

10

15

20

25

30

35

40

1 5 9 13 17

NCC

21 25 29 33 37 41 45 49 53 57 61 65 69 73 77 81 85 89 93 97

TOPSIS-REEBSAMRandom

The number of nodes to be attacked

Figure 11 NCC of Shanxi railway network

Complexity 11

method +erefore EBSAM performs better than othermethods

6 Conclusions and Future Work

In this paper we proposed an entropy theory-based self-adaptive node importance evaluation method for complexnetworks Firstly we select four centrality measures whichcan reflect different characteristics of the node as nodeimportance evaluation indicators +en we combine themtogether with appropriate weights calculated by an entropytheory-based algorithm +e algorithm shows a strongadaptability and thus allows be widely implemented indifferent kinds of networks In the traditional method theweights are selected based on the subjective experience of theresearchers instead of enough scientific basis which wouldlead to inaccurate evaluation results +e proposed methodis better because it utilizes entropy theory to calculate theweight of each indicator A smaller value of entropy indicatesthat the corresponding attribution contains less useful in-formation In a multiattribute decision-making problem weneed to assign a larger weight to attribute with more usefulinformation rather than the attribute with greater uncer-tainty So with this algorithm we can better assign properweight to different attributions +e experimental results onthree types of real-world complex networks show that ourmethod performs better with compared methods Our on-going research will focus on investigating the effectiveness ofour method in more complex environments

Data Availability

+e data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

+e authors declare that they have no conflicts of interest

Acknowledgments

+is study was supported by National Key RampD Program ofChina (Funding No 2018YFB1402800) and the NaturalScience Foundation of China (No 61571066)

References

[1] SWang X Zhang Y Zhang LWang J Yang andWWangldquoA survey on mobile edge networks convergence of com-puting caching and communicationsrdquo Ieee Access vol 5pp 6757ndash6779 2017 (in English)

[2] X Ren and L Lu ldquoReview of ranking nodes in complexnetworksrdquo Chinese Science Bulletin vol 59 no 13pp 1175ndash1197 2014

[3] K L S Samanta and M Pal ldquoStudy on centrality measures insocial networks a surveyrdquo Social Network Analysis Miningvol 8 no 1 p 13 2018

[4] J Dong F Ye W Chen and J Wu ldquoIdentifying influentialnodes in complex networks via semi-local centralityrdquo inProceedings of the 2018 IEEE International Symposium on

Circuits and Systems (Iscas) (in English) Florence Italy May2018

[5] K Angelou M Maragakis and P Argyrakis ldquoA structuralanalysis of the patent citation network by the k-shell de-composition methodrdquo Physica A-Statistical Mechanics and ItsApplications vol 521 pp 476ndash483 2019 (in English)

[6] G Li M Li J Wang Y Li and Y Pan ldquoUnited neighborhoodcloseness centrality and orthology for predicting essentialproteinsrdquo IEEEACM Transactions on Computational Biologyand Bioinformatics 2018

[7] M Riondato and E Upfal ldquoABRA approximating be-tweenness centrality in static and dynamic graphs withrademacher averagesrdquo ACM Transactions on KnowledgeDiscovery from Data vol 12 no 5 pp 1ndash38 2018 (inEnglish)

[8] N Gupta A Singh and H Cherifi ldquoCentrality measures fornetworks with community structurerdquo Physica A StatisticalMechanics and Its Applications vol 452 pp 46ndash59 2016 (inEnglish)

[9] D Chen H Gao L Lu and T Zhou ldquoIdentifying influentialnodes in large-scale directed networks the role of clusteringrdquoPLoS One vol 8 Article ID e77455 no 10 (in English) 2013

[10] K Tu P Cui X Wang P Yu and W Zhu ldquoDeep recursivenetwork embedding with regular equivalencerdquo in Proceedingsof the Kddrsquo18 24th Acm Sigkdd International Conference onKnowledge Discovery amp Data Mining London UKpp 2357ndash2366 August 2018 (in English)

[11] R Poulin M-C Boily and B R Masse ldquoDynamical systemsto define centrality in social networksrdquo Social Networksvol 22 no 3 pp 187ndash220 2000 (in English)

[12] P Hu and T Mei ldquoRanking influential nodes in complexnetworks with structural holesrdquo Physica A Statistical Me-chanics and Its Applications vol 490 pp 624ndash631 15 2018 (inEnglish)

[13] L Lu Y Zhang C Yeung and T Zhou ldquoLeaders in socialnetworks the delicious caserdquo Plos One vol 6 no 6 (inEnglish) 2011

[14] J Liu Q Xiong W Shi X Shi and K Wang ldquoEvaluating theimportance of nodes in complex networksrdquo Physica A Sta-tistical Mechanics and Its Applications vol 452 pp 209ndash2192016 (in English)

[15] F Hu Y Liu and J Jin ldquoMulti-index evaluation algorithmbased on locally linear embedding for the node importance incomplex networksrdquo in Proceedings of ampirteenth InternationalSymposium on Distributed Computing and Applications toBusiness Engineering and Science (Dcabes 2014) pp 138ndash142(in English) Xian Ning China November 2014

[16] S T Roweis and L K Saul ldquoNonlinear dimensionality re-duction by locally linear embeddingrdquo Science vol 290no 5500 pp 2323ndash2326 2000 (in English)

[17] L Xu J Liu Y Liu Y Liu J Gou and B Masoud ldquoNodeimportance classified comprehensive assessmentrdquo Proceed-ings of the Csee vol 34 no 10 pp 1609ndash1617 2014

[18] X Zhang and X Wang ldquoEvaluation formula for communi-cation network node importancerdquo Journal of NortheasternUniversity vol 35 no 5 pp 663ndash666 2014

[19] H Ping W Fan and S Mei ldquoIdentifying node importance incomplex networksrdquo Physica A Statistical Mechanics and ItsApplications vol 429 pp 169ndash176 2015

[20] Y Yu ldquoNode importance measurement based on the degreeand closeness centralityrdquo Journal of Information and Com-putational Science vol 12 no 3 pp 1281ndash1291 2015

12 Complexity

[21] Q Hu Y Yin P Ma Y Gao Y Zhang and C Xing ldquoA newapproach to identify influential spreaders in complex net-worksrdquoActa Physica Sinica vol 62 no 14 (in Chinese) 2013

[22] Y Zhang Y Bao S Zhao J Chen and J Tang ldquoIdentifyingnode importance by combining betweenness centrality andKatz centralityrdquo in Proceedings of the 2015 InternationalConference on Cloud Computing and Big Data (Ccbd)pp 354ndash357 (in English) Shanghai China November 2015

[23] Y Yang and G Xie ldquoEfficient identification of node im-portance in social networksrdquo Information Processing ampManagement vol 52 no 5 pp 911ndash922 2016 (in English)

[24] Z Liu C Jiang J Wang and H Yu ldquo+e node importance inactual complex networks based on a multi-attribute rankingmethodrdquo Knowledge-Based Systems vol 84 pp 56ndash66 2015(in English)

[25] C E Shannon ldquoA mathematical theory of communicationrdquoBell System Technical Journal vol 27 no 4 pp 623ndash656 1948

[26] Q Zhang M Li and Y Deng ldquoMeasure the structure sim-ilarity of nodes in complex networks based on relative en-tropyrdquo Physica A Statistical Mechanics and Its Applicationsvol 491 pp 749ndash763 2018 (in English)

[27] Y He H Guo M Jin and P Ren ldquoA linguistic entropyweight method and its application in linguistic multi-attributegroup decision makingrdquo Nonlinear Dynamics vol 84 no 1pp 399ndash404 2016 (in English)

[28] X Liang and C Wei ldquoAn Atanassovrsquos intuitionistic fuzzymulti-attribute group decision making method based onentropy and similarity measurerdquo International Journal ofMachine Learning and Cybernetics vol 5 no 3 pp 435ndash4442014 (in English)

[29] T-Y Chen and C-H Li ldquoObjective weights with intuition-istic fuzzy entropy measures and computational experimentanalysisrdquo Applied Soft Computing vol 11 no 8 pp 5411ndash5423 2011 (in English)

[30] S-P Wan Q-Y Wang and J-Y Dong ldquo+e extendedVIKOR method for multi-attribute group decision makingwith triangular intuitionistic fuzzy numbersrdquo Knowledge-Based Systems vol 52 pp 65ndash77 2013 (in English)

[31] H Nguyen ldquoA new knowledge-based measure for intui-tionistic fuzzy sets and its application in multiple attributegroup decision makingrdquo Expert Systems with Applicationsvol 42 no 22 pp 8766ndash8774 2015 (in English)

[32] S Dereich and P Morters ldquoRandom networks with sublinearpreferential attachment the giant componentrdquoampe Annals ofProbability vol 41 no 1 pp 329ndash384 2013 (in English)

[33] C M Schneider A A Moreira J S Andrade S Havlin andH J Herrmann ldquoMitigation of malicious attacks on net-worksrdquo Proceedings of the National Academy of Sciences of theUnited States of America vol 108 no 10 pp 3838ndash3841 2011(in English)

[34] A Zareie and A Sheikhahmadi ldquoA hierarchical approach forinfluential node ranking in complex social networksrdquo ExpertSystems with Applications vol 93 pp 200ndash211 2018 (inEnglish)

Complexity 13

523 Experimental Results on Shanxi Railway NetworkFinally we conduct experiment on Shanxi Railway Net-work As shown in Figure 9 Shanxi Railway Network is apart of the transportation network in Shanxi It providesgreat convenience for peoplersquos outgoing and commodities

trading +e topological structure of Shanxi Water Net-work is shown in Figure 10 +e experimental result isshown in Figure 11 +e network is coming to break downafter the top 60 nodes have been attacked +e NCC ofShanxi Water Network obtains the largest ascent with our

4

31811

1

2

13

1219

2123

22

20

9

24

31

6

25

10

87

5

26

14

28

16 17

15

29

93

85

90

96

91

92

84

94

9597

98

33

27

30

3435

43

5150

3236

45

44

58

4138

40

3937

42

81

6465 66

7982

80 78

76

77

83

53

52

59

54

60

55

61

4948

46

47

86

62

87

63

57

89

56

886968

71

73 72

70

7475

67

Figure 10 Topological structure of Shanxi railway network

0

5

10

15

20

25

30

35

40

1 5 9 13 17

NCC

21 25 29 33 37 41 45 49 53 57 61 65 69 73 77 81 85 89 93 97

TOPSIS-REEBSAMRandom

The number of nodes to be attacked

Figure 11 NCC of Shanxi railway network

Complexity 11

method +erefore EBSAM performs better than othermethods

6 Conclusions and Future Work

In this paper we proposed an entropy theory-based self-adaptive node importance evaluation method for complexnetworks Firstly we select four centrality measures whichcan reflect different characteristics of the node as nodeimportance evaluation indicators +en we combine themtogether with appropriate weights calculated by an entropytheory-based algorithm +e algorithm shows a strongadaptability and thus allows be widely implemented indifferent kinds of networks In the traditional method theweights are selected based on the subjective experience of theresearchers instead of enough scientific basis which wouldlead to inaccurate evaluation results +e proposed methodis better because it utilizes entropy theory to calculate theweight of each indicator A smaller value of entropy indicatesthat the corresponding attribution contains less useful in-formation In a multiattribute decision-making problem weneed to assign a larger weight to attribute with more usefulinformation rather than the attribute with greater uncer-tainty So with this algorithm we can better assign properweight to different attributions +e experimental results onthree types of real-world complex networks show that ourmethod performs better with compared methods Our on-going research will focus on investigating the effectiveness ofour method in more complex environments

Data Availability

+e data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

+e authors declare that they have no conflicts of interest

Acknowledgments

+is study was supported by National Key RampD Program ofChina (Funding No 2018YFB1402800) and the NaturalScience Foundation of China (No 61571066)

References

[1] SWang X Zhang Y Zhang LWang J Yang andWWangldquoA survey on mobile edge networks convergence of com-puting caching and communicationsrdquo Ieee Access vol 5pp 6757ndash6779 2017 (in English)

[2] X Ren and L Lu ldquoReview of ranking nodes in complexnetworksrdquo Chinese Science Bulletin vol 59 no 13pp 1175ndash1197 2014

[3] K L S Samanta and M Pal ldquoStudy on centrality measures insocial networks a surveyrdquo Social Network Analysis Miningvol 8 no 1 p 13 2018

[4] J Dong F Ye W Chen and J Wu ldquoIdentifying influentialnodes in complex networks via semi-local centralityrdquo inProceedings of the 2018 IEEE International Symposium on

Circuits and Systems (Iscas) (in English) Florence Italy May2018

[5] K Angelou M Maragakis and P Argyrakis ldquoA structuralanalysis of the patent citation network by the k-shell de-composition methodrdquo Physica A-Statistical Mechanics and ItsApplications vol 521 pp 476ndash483 2019 (in English)

[6] G Li M Li J Wang Y Li and Y Pan ldquoUnited neighborhoodcloseness centrality and orthology for predicting essentialproteinsrdquo IEEEACM Transactions on Computational Biologyand Bioinformatics 2018

[7] M Riondato and E Upfal ldquoABRA approximating be-tweenness centrality in static and dynamic graphs withrademacher averagesrdquo ACM Transactions on KnowledgeDiscovery from Data vol 12 no 5 pp 1ndash38 2018 (inEnglish)

[8] N Gupta A Singh and H Cherifi ldquoCentrality measures fornetworks with community structurerdquo Physica A StatisticalMechanics and Its Applications vol 452 pp 46ndash59 2016 (inEnglish)

[9] D Chen H Gao L Lu and T Zhou ldquoIdentifying influentialnodes in large-scale directed networks the role of clusteringrdquoPLoS One vol 8 Article ID e77455 no 10 (in English) 2013

[10] K Tu P Cui X Wang P Yu and W Zhu ldquoDeep recursivenetwork embedding with regular equivalencerdquo in Proceedingsof the Kddrsquo18 24th Acm Sigkdd International Conference onKnowledge Discovery amp Data Mining London UKpp 2357ndash2366 August 2018 (in English)

[11] R Poulin M-C Boily and B R Masse ldquoDynamical systemsto define centrality in social networksrdquo Social Networksvol 22 no 3 pp 187ndash220 2000 (in English)

[12] P Hu and T Mei ldquoRanking influential nodes in complexnetworks with structural holesrdquo Physica A Statistical Me-chanics and Its Applications vol 490 pp 624ndash631 15 2018 (inEnglish)

[13] L Lu Y Zhang C Yeung and T Zhou ldquoLeaders in socialnetworks the delicious caserdquo Plos One vol 6 no 6 (inEnglish) 2011

[14] J Liu Q Xiong W Shi X Shi and K Wang ldquoEvaluating theimportance of nodes in complex networksrdquo Physica A Sta-tistical Mechanics and Its Applications vol 452 pp 209ndash2192016 (in English)

[15] F Hu Y Liu and J Jin ldquoMulti-index evaluation algorithmbased on locally linear embedding for the node importance incomplex networksrdquo in Proceedings of ampirteenth InternationalSymposium on Distributed Computing and Applications toBusiness Engineering and Science (Dcabes 2014) pp 138ndash142(in English) Xian Ning China November 2014

[16] S T Roweis and L K Saul ldquoNonlinear dimensionality re-duction by locally linear embeddingrdquo Science vol 290no 5500 pp 2323ndash2326 2000 (in English)

[17] L Xu J Liu Y Liu Y Liu J Gou and B Masoud ldquoNodeimportance classified comprehensive assessmentrdquo Proceed-ings of the Csee vol 34 no 10 pp 1609ndash1617 2014

[18] X Zhang and X Wang ldquoEvaluation formula for communi-cation network node importancerdquo Journal of NortheasternUniversity vol 35 no 5 pp 663ndash666 2014

[19] H Ping W Fan and S Mei ldquoIdentifying node importance incomplex networksrdquo Physica A Statistical Mechanics and ItsApplications vol 429 pp 169ndash176 2015

[20] Y Yu ldquoNode importance measurement based on the degreeand closeness centralityrdquo Journal of Information and Com-putational Science vol 12 no 3 pp 1281ndash1291 2015

12 Complexity

[21] Q Hu Y Yin P Ma Y Gao Y Zhang and C Xing ldquoA newapproach to identify influential spreaders in complex net-worksrdquoActa Physica Sinica vol 62 no 14 (in Chinese) 2013

[22] Y Zhang Y Bao S Zhao J Chen and J Tang ldquoIdentifyingnode importance by combining betweenness centrality andKatz centralityrdquo in Proceedings of the 2015 InternationalConference on Cloud Computing and Big Data (Ccbd)pp 354ndash357 (in English) Shanghai China November 2015

[23] Y Yang and G Xie ldquoEfficient identification of node im-portance in social networksrdquo Information Processing ampManagement vol 52 no 5 pp 911ndash922 2016 (in English)

[24] Z Liu C Jiang J Wang and H Yu ldquo+e node importance inactual complex networks based on a multi-attribute rankingmethodrdquo Knowledge-Based Systems vol 84 pp 56ndash66 2015(in English)

[25] C E Shannon ldquoA mathematical theory of communicationrdquoBell System Technical Journal vol 27 no 4 pp 623ndash656 1948

[26] Q Zhang M Li and Y Deng ldquoMeasure the structure sim-ilarity of nodes in complex networks based on relative en-tropyrdquo Physica A Statistical Mechanics and Its Applicationsvol 491 pp 749ndash763 2018 (in English)

[27] Y He H Guo M Jin and P Ren ldquoA linguistic entropyweight method and its application in linguistic multi-attributegroup decision makingrdquo Nonlinear Dynamics vol 84 no 1pp 399ndash404 2016 (in English)

[28] X Liang and C Wei ldquoAn Atanassovrsquos intuitionistic fuzzymulti-attribute group decision making method based onentropy and similarity measurerdquo International Journal ofMachine Learning and Cybernetics vol 5 no 3 pp 435ndash4442014 (in English)

[29] T-Y Chen and C-H Li ldquoObjective weights with intuition-istic fuzzy entropy measures and computational experimentanalysisrdquo Applied Soft Computing vol 11 no 8 pp 5411ndash5423 2011 (in English)

[30] S-P Wan Q-Y Wang and J-Y Dong ldquo+e extendedVIKOR method for multi-attribute group decision makingwith triangular intuitionistic fuzzy numbersrdquo Knowledge-Based Systems vol 52 pp 65ndash77 2013 (in English)

[31] H Nguyen ldquoA new knowledge-based measure for intui-tionistic fuzzy sets and its application in multiple attributegroup decision makingrdquo Expert Systems with Applicationsvol 42 no 22 pp 8766ndash8774 2015 (in English)

[32] S Dereich and P Morters ldquoRandom networks with sublinearpreferential attachment the giant componentrdquoampe Annals ofProbability vol 41 no 1 pp 329ndash384 2013 (in English)

[33] C M Schneider A A Moreira J S Andrade S Havlin andH J Herrmann ldquoMitigation of malicious attacks on net-worksrdquo Proceedings of the National Academy of Sciences of theUnited States of America vol 108 no 10 pp 3838ndash3841 2011(in English)

[34] A Zareie and A Sheikhahmadi ldquoA hierarchical approach forinfluential node ranking in complex social networksrdquo ExpertSystems with Applications vol 93 pp 200ndash211 2018 (inEnglish)

Complexity 13

method +erefore EBSAM performs better than othermethods

6 Conclusions and Future Work

In this paper we proposed an entropy theory-based self-adaptive node importance evaluation method for complexnetworks Firstly we select four centrality measures whichcan reflect different characteristics of the node as nodeimportance evaluation indicators +en we combine themtogether with appropriate weights calculated by an entropytheory-based algorithm +e algorithm shows a strongadaptability and thus allows be widely implemented indifferent kinds of networks In the traditional method theweights are selected based on the subjective experience of theresearchers instead of enough scientific basis which wouldlead to inaccurate evaluation results +e proposed methodis better because it utilizes entropy theory to calculate theweight of each indicator A smaller value of entropy indicatesthat the corresponding attribution contains less useful in-formation In a multiattribute decision-making problem weneed to assign a larger weight to attribute with more usefulinformation rather than the attribute with greater uncer-tainty So with this algorithm we can better assign properweight to different attributions +e experimental results onthree types of real-world complex networks show that ourmethod performs better with compared methods Our on-going research will focus on investigating the effectiveness ofour method in more complex environments

Data Availability

+e data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

+e authors declare that they have no conflicts of interest

Acknowledgments

+is study was supported by National Key RampD Program ofChina (Funding No 2018YFB1402800) and the NaturalScience Foundation of China (No 61571066)

References

[1] SWang X Zhang Y Zhang LWang J Yang andWWangldquoA survey on mobile edge networks convergence of com-puting caching and communicationsrdquo Ieee Access vol 5pp 6757ndash6779 2017 (in English)

[2] X Ren and L Lu ldquoReview of ranking nodes in complexnetworksrdquo Chinese Science Bulletin vol 59 no 13pp 1175ndash1197 2014

[3] K L S Samanta and M Pal ldquoStudy on centrality measures insocial networks a surveyrdquo Social Network Analysis Miningvol 8 no 1 p 13 2018

[4] J Dong F Ye W Chen and J Wu ldquoIdentifying influentialnodes in complex networks via semi-local centralityrdquo inProceedings of the 2018 IEEE International Symposium on

Circuits and Systems (Iscas) (in English) Florence Italy May2018

[5] K Angelou M Maragakis and P Argyrakis ldquoA structuralanalysis of the patent citation network by the k-shell de-composition methodrdquo Physica A-Statistical Mechanics and ItsApplications vol 521 pp 476ndash483 2019 (in English)

[6] G Li M Li J Wang Y Li and Y Pan ldquoUnited neighborhoodcloseness centrality and orthology for predicting essentialproteinsrdquo IEEEACM Transactions on Computational Biologyand Bioinformatics 2018

[7] M Riondato and E Upfal ldquoABRA approximating be-tweenness centrality in static and dynamic graphs withrademacher averagesrdquo ACM Transactions on KnowledgeDiscovery from Data vol 12 no 5 pp 1ndash38 2018 (inEnglish)

[8] N Gupta A Singh and H Cherifi ldquoCentrality measures fornetworks with community structurerdquo Physica A StatisticalMechanics and Its Applications vol 452 pp 46ndash59 2016 (inEnglish)

[9] D Chen H Gao L Lu and T Zhou ldquoIdentifying influentialnodes in large-scale directed networks the role of clusteringrdquoPLoS One vol 8 Article ID e77455 no 10 (in English) 2013

[10] K Tu P Cui X Wang P Yu and W Zhu ldquoDeep recursivenetwork embedding with regular equivalencerdquo in Proceedingsof the Kddrsquo18 24th Acm Sigkdd International Conference onKnowledge Discovery amp Data Mining London UKpp 2357ndash2366 August 2018 (in English)

[11] R Poulin M-C Boily and B R Masse ldquoDynamical systemsto define centrality in social networksrdquo Social Networksvol 22 no 3 pp 187ndash220 2000 (in English)

[12] P Hu and T Mei ldquoRanking influential nodes in complexnetworks with structural holesrdquo Physica A Statistical Me-chanics and Its Applications vol 490 pp 624ndash631 15 2018 (inEnglish)

[13] L Lu Y Zhang C Yeung and T Zhou ldquoLeaders in socialnetworks the delicious caserdquo Plos One vol 6 no 6 (inEnglish) 2011

[14] J Liu Q Xiong W Shi X Shi and K Wang ldquoEvaluating theimportance of nodes in complex networksrdquo Physica A Sta-tistical Mechanics and Its Applications vol 452 pp 209ndash2192016 (in English)

[15] F Hu Y Liu and J Jin ldquoMulti-index evaluation algorithmbased on locally linear embedding for the node importance incomplex networksrdquo in Proceedings of ampirteenth InternationalSymposium on Distributed Computing and Applications toBusiness Engineering and Science (Dcabes 2014) pp 138ndash142(in English) Xian Ning China November 2014

[16] S T Roweis and L K Saul ldquoNonlinear dimensionality re-duction by locally linear embeddingrdquo Science vol 290no 5500 pp 2323ndash2326 2000 (in English)

[17] L Xu J Liu Y Liu Y Liu J Gou and B Masoud ldquoNodeimportance classified comprehensive assessmentrdquo Proceed-ings of the Csee vol 34 no 10 pp 1609ndash1617 2014

[18] X Zhang and X Wang ldquoEvaluation formula for communi-cation network node importancerdquo Journal of NortheasternUniversity vol 35 no 5 pp 663ndash666 2014

[19] H Ping W Fan and S Mei ldquoIdentifying node importance incomplex networksrdquo Physica A Statistical Mechanics and ItsApplications vol 429 pp 169ndash176 2015

[20] Y Yu ldquoNode importance measurement based on the degreeand closeness centralityrdquo Journal of Information and Com-putational Science vol 12 no 3 pp 1281ndash1291 2015

12 Complexity

[21] Q Hu Y Yin P Ma Y Gao Y Zhang and C Xing ldquoA newapproach to identify influential spreaders in complex net-worksrdquoActa Physica Sinica vol 62 no 14 (in Chinese) 2013

[22] Y Zhang Y Bao S Zhao J Chen and J Tang ldquoIdentifyingnode importance by combining betweenness centrality andKatz centralityrdquo in Proceedings of the 2015 InternationalConference on Cloud Computing and Big Data (Ccbd)pp 354ndash357 (in English) Shanghai China November 2015

[23] Y Yang and G Xie ldquoEfficient identification of node im-portance in social networksrdquo Information Processing ampManagement vol 52 no 5 pp 911ndash922 2016 (in English)

[24] Z Liu C Jiang J Wang and H Yu ldquo+e node importance inactual complex networks based on a multi-attribute rankingmethodrdquo Knowledge-Based Systems vol 84 pp 56ndash66 2015(in English)

[25] C E Shannon ldquoA mathematical theory of communicationrdquoBell System Technical Journal vol 27 no 4 pp 623ndash656 1948

[26] Q Zhang M Li and Y Deng ldquoMeasure the structure sim-ilarity of nodes in complex networks based on relative en-tropyrdquo Physica A Statistical Mechanics and Its Applicationsvol 491 pp 749ndash763 2018 (in English)

[27] Y He H Guo M Jin and P Ren ldquoA linguistic entropyweight method and its application in linguistic multi-attributegroup decision makingrdquo Nonlinear Dynamics vol 84 no 1pp 399ndash404 2016 (in English)

[28] X Liang and C Wei ldquoAn Atanassovrsquos intuitionistic fuzzymulti-attribute group decision making method based onentropy and similarity measurerdquo International Journal ofMachine Learning and Cybernetics vol 5 no 3 pp 435ndash4442014 (in English)

[29] T-Y Chen and C-H Li ldquoObjective weights with intuition-istic fuzzy entropy measures and computational experimentanalysisrdquo Applied Soft Computing vol 11 no 8 pp 5411ndash5423 2011 (in English)

[30] S-P Wan Q-Y Wang and J-Y Dong ldquo+e extendedVIKOR method for multi-attribute group decision makingwith triangular intuitionistic fuzzy numbersrdquo Knowledge-Based Systems vol 52 pp 65ndash77 2013 (in English)

[31] H Nguyen ldquoA new knowledge-based measure for intui-tionistic fuzzy sets and its application in multiple attributegroup decision makingrdquo Expert Systems with Applicationsvol 42 no 22 pp 8766ndash8774 2015 (in English)

[32] S Dereich and P Morters ldquoRandom networks with sublinearpreferential attachment the giant componentrdquoampe Annals ofProbability vol 41 no 1 pp 329ndash384 2013 (in English)

[33] C M Schneider A A Moreira J S Andrade S Havlin andH J Herrmann ldquoMitigation of malicious attacks on net-worksrdquo Proceedings of the National Academy of Sciences of theUnited States of America vol 108 no 10 pp 3838ndash3841 2011(in English)

[34] A Zareie and A Sheikhahmadi ldquoA hierarchical approach forinfluential node ranking in complex social networksrdquo ExpertSystems with Applications vol 93 pp 200ndash211 2018 (inEnglish)

Complexity 13

[21] Q Hu Y Yin P Ma Y Gao Y Zhang and C Xing ldquoA newapproach to identify influential spreaders in complex net-worksrdquoActa Physica Sinica vol 62 no 14 (in Chinese) 2013

[22] Y Zhang Y Bao S Zhao J Chen and J Tang ldquoIdentifyingnode importance by combining betweenness centrality andKatz centralityrdquo in Proceedings of the 2015 InternationalConference on Cloud Computing and Big Data (Ccbd)pp 354ndash357 (in English) Shanghai China November 2015

[23] Y Yang and G Xie ldquoEfficient identification of node im-portance in social networksrdquo Information Processing ampManagement vol 52 no 5 pp 911ndash922 2016 (in English)

[24] Z Liu C Jiang J Wang and H Yu ldquo+e node importance inactual complex networks based on a multi-attribute rankingmethodrdquo Knowledge-Based Systems vol 84 pp 56ndash66 2015(in English)

[25] C E Shannon ldquoA mathematical theory of communicationrdquoBell System Technical Journal vol 27 no 4 pp 623ndash656 1948

[26] Q Zhang M Li and Y Deng ldquoMeasure the structure sim-ilarity of nodes in complex networks based on relative en-tropyrdquo Physica A Statistical Mechanics and Its Applicationsvol 491 pp 749ndash763 2018 (in English)

[27] Y He H Guo M Jin and P Ren ldquoA linguistic entropyweight method and its application in linguistic multi-attributegroup decision makingrdquo Nonlinear Dynamics vol 84 no 1pp 399ndash404 2016 (in English)

[28] X Liang and C Wei ldquoAn Atanassovrsquos intuitionistic fuzzymulti-attribute group decision making method based onentropy and similarity measurerdquo International Journal ofMachine Learning and Cybernetics vol 5 no 3 pp 435ndash4442014 (in English)

[29] T-Y Chen and C-H Li ldquoObjective weights with intuition-istic fuzzy entropy measures and computational experimentanalysisrdquo Applied Soft Computing vol 11 no 8 pp 5411ndash5423 2011 (in English)

[30] S-P Wan Q-Y Wang and J-Y Dong ldquo+e extendedVIKOR method for multi-attribute group decision makingwith triangular intuitionistic fuzzy numbersrdquo Knowledge-Based Systems vol 52 pp 65ndash77 2013 (in English)

[31] H Nguyen ldquoA new knowledge-based measure for intui-tionistic fuzzy sets and its application in multiple attributegroup decision makingrdquo Expert Systems with Applicationsvol 42 no 22 pp 8766ndash8774 2015 (in English)

[32] S Dereich and P Morters ldquoRandom networks with sublinearpreferential attachment the giant componentrdquoampe Annals ofProbability vol 41 no 1 pp 329ndash384 2013 (in English)

[33] C M Schneider A A Moreira J S Andrade S Havlin andH J Herrmann ldquoMitigation of malicious attacks on net-worksrdquo Proceedings of the National Academy of Sciences of theUnited States of America vol 108 no 10 pp 3838ndash3841 2011(in English)

[34] A Zareie and A Sheikhahmadi ldquoA hierarchical approach forinfluential node ranking in complex social networksrdquo ExpertSystems with Applications vol 93 pp 200ndash211 2018 (inEnglish)

Complexity 13