TRANSPORT ON COMPLEX NETWORKS: FLOW, JAMMING AND OPTIMIZATIONtadic/pdfs/review_IJBCH.pdf · tered...

International Journal of Bifurcation and Chaos, Vol. 17, No. 7 (2007) 2363–2385c© World Scientific Publishing Company

TRANSPORT ON COMPLEX NETWORKS: FLOW,JAMMING AND OPTIMIZATION

BOSILJKA TADICDepartment for Theoretical Physics, Jozef Stefan Institute,

P. O. Box 3000, 1001 Ljubljana, Slovenia

G. J. RODGERSDepartment of Mathematical Sciences, Brunel University,

Uxbridge, Middlesex UB8 3PH, UK

STEFAN THURNERComplex Systems Research Group, Medical University of Vienna HNO,

Wahringer Gurtel 18-20, A-1090 Vienna, Austria

Received April 12, 2006; Revised July 3, 2006

Many transport processes on networks depend crucially on the underlying network geometry,although the exact relationship between the structure of the network and the properties oftransport processes remain elusive. In this paper, we address this question by using numericalmodels in which both structure and dynamics are controlled systematically. We consider thetraffic of information packets that include driving, searching and queuing. We present the resultsof extensive simulations on two classes of networks; a correlated cyclic scale-free network and anuncorrelated homogeneous weakly clustered network. By measuring different dynamical variablesin the free flow regime we show how the global statistical properties of the transport are relatedto the temporal fluctuations at individual nodes (the traffic noise) and the links (the trafficflow). We then demonstrate that these two network classes appear as representative topologiesfor optimal traffic flow in the regimes of low density and high density traffic, respectively. We alsodetermine statistical indicators of the pre-jamming regime on different network geometries anddiscuss the role of queuing and dynamical betweenness for the traffic congestion. The transitionto the jammed traffic regime at a critical posting rate on different network topologies is studied asa phase transition with an appropriate order parameter. We also address several open theoreticalproblems related to the network dynamics.

Keywords : Information traffic; correlated scale-free networks; jamming indicators; dynamicbetweenness.

1. Introduction

1.1. Motivation

Networks form part of the basic model for virtu-ally every known co-operative phenomenon, andthe study of processes on networks is funda-mental to a large part of the physical, biolog-ical, social, economic and engineering sciences[Boccaletti et al., 2006]. Most technological, bio-logical, economical or social networks support a

number of transport processes, such as the traf-fic of information packets [Tadic & Rodgers, 2002;Tadic & Thurner, 2004; Tadic et al., 2004; Arenaset al., 2001; Guimera et al., 2002a; Sole & Valverde,2001; Valverde & Sole, 2002; Ohira & Sawatari,1998; Moreno et al., 2003; Rosvall et al., 2005;Rosvall & Sneppen, 2003; Valverde & Sole, 2004;Wang et al., 2006], signals, molecules [Agrawal,2002], finance and wealth [Thurner et al., 2003;

2363

2364 B. Tadic et al.

Boss et al., 2004; Coelho et al., 2005], rumors[Moreno et al., 2004] or diseases [Newman, 2002;Boguna et al., 2003]. One can model these processesin many different ways, from simple interacting ran-dom walkers [Tadic, 2001a; Noh & Rieger, 2004;Eisler & Kertesz, 2005], to interacting random walkswith random traps [Gallos, 2004] on the networkthat serve as destinations, to richer models in whichthe traps are walker specific. Some of the modelswe discuss here allow an even wider range of behav-ior; each walker has a destination which is knownand fixed at the start, potentially allowing bothlocal and global search and navigation strategiesto be introduced to improve the efficiency of thetransport.

In recent years, studies of different networkshave revealed that the dynamic processes cruciallydepend on the topology of the underlying networkstructure [Tadic et al., 2004; Guimera et al., 2002a].However, the precise interdependencies between thefunctional properties and the relevant structuralparameters, which define network classes, remainlargely unknown. In naturally evolved networks,such as the cytoskeleton in cells, which serve asa backbone of molecular transport within the cell,efficient functioning emerges through an optimiza-tion of function via the evolutionary adjustmentof the structure. This structure — function opti-mization — is an imperative in artificial networksbecause of the need for increased efficiency [Latora& Marchiori, 2001; Jarrett et al., 2005], low risk ofinformation loss, low levels of congestion [Guimeraet al., 2002b; Moreno et al., 2003] and low risk ofcritical malfunction [Holme, 2003; Echenique et al.,2005; Ashton et al., 2005]. Therefore, understandinga network’s functional properties and their depen-dence on a reduced set of its structural parame-ters is of paramount importance. This understand-ing is being developed through the systematic col-lection of empirical data, and through the intro-duction, simulation and solution of new algorithmsthat seek to improve one or more aspects of thefunctional performance of the network. This workprovides the basis for a deeper theoretical under-standing of transport on networks, that allows theidentification of universality classes of both networktopologies and transport processes.

1.2. Methodology

This paper reviews some of the work that has beencarried out to progress this agenda. In our approach

both the network structure and the dynamics onthe network are controlled systematically withina numerical model of information traffic. In thismodel the posting, navigation and queuing ofpackets are parameterized by a set of control param-eters. Then the properties of the traffic are deter-mined for different underlying network structures.Two approaches are available to optimize traffic.When the network structure is fixed on the timescale of the transport, one can find sets of optimalpaths between nodes and specify the traffic alongthese paths. Alternatively, when the network struc-ture is allowed to evolve on the same time scaleas the transport, an optimal network structure isfound that depends on the traffic conditions.

1.3. Properties of informationtraffic

The traffic of information packets is defined via sev-eral properties and parameters. Here we summarizethese properties, which are later realized in the sim-ulations.

• Creation & Assignment. An information packet iscreated, at some rate, at a node i and is assignedthe address of another node j where it should bedelivered; When a packet arrives at its destina-tion it is removed from the network;

• Traffic Queues. Packets form queues when two ormore packets are at same node at the same time;The queue length at node i at time t is denotedby Qi(t);

• Queuing Discipline. This determines the order inwhich packets leave a queue; In this work we usethe last-in-first-out (LIFO) queuing discipline fortraffic simulations;

• Waiting Time tw. Is the time that a packet spendsin a particular queue waiting to leave;

• Travel Time T . Is the total time that a packetspends in the network from its creation to itsdelivery at its destination. This time is equal tothe summation of the path length and the wait-ing time at each queue along the path; The traveltime of a packet is related to its travel costs, some-times called the search costs.

• Navigation & Search Depth. Information abouta packet’s destination is required if one wishesto reduce its travel time by using a navigationprocess; Global navigation is a costly procedurein which a shortest (or the best) path for eachpacket is determined; A much less costly alterna-tive is to use local search algorithms in which each

Transport on Complex Networks: Flow, Jamming and Optimization 2365

packet explores the neighborhood of its currentnode for its destination address or for an optimumdirection; A search depth of d = 1 corresponds tothe nearest-neighborhood of a node and d = 2corresponds to the next-nearest-neighborhood ofa node, sometimes called nnn-search; A searchdepth of d = 0 corresponds to random process-ing; In this paper we employ d = 2, nnn-search;

• Dynamic Load. The number of packets trans-ported by a node i, or along a link ij, at timet defines the dynamic load of the node hi(t) andthe link fij(t); The total number of packets inthe network Np(t) is the network load, obviouslyNp(t) =

∑i Qi(t), the sum of all queue lengths at

time t;• Traffic Noise. The time signals of temporally

fluctuating variables such as hi(t), Np(t), thenumber of simultaneously active nodes n(t), etc.are often called traffic noise signals. In particu-lar, the set of signals {hi(t)} recorded at eachof i = 1, 2, . . . , N nodes simultaneously formthe basis for multichannel noise analysis; Anal-ogously, traffic flow is the same quantity definedfor links rather than nodes;

• Free Flow Regime. Refers to the free (uncon-gested) flow of packets, which is compatible withstationarity of the traffic noise time-series;

• Congested Regime. Corresponds to a partial orcomplete jamming in networks when packets canget stuck for an indefinite time; The network loadincreases steadily with time, making the time-series non-stationary.

1.4. Concepts

We investigate our numerical model for informationpacket transport on two types of networks; a clus-tered scale-free network and a homogeneous net-work, which are, respectively, representatives of thecausal and homogeneous [Burda et al., 2004] net-work classes. The topological properties of thesenetworks that are relevant to transport process arediscussed in detail in Sec. 2, along with the trans-port rules. We then present the results of simu-lations of traffic on these networks. These resultsconsist of statistical properties of traffic collectedon the global network level (such as probabilitydistributions of packet travel times and waitingtime, etc.), and local (individual nodes and links)activity during transport, sometimes called multi-channel noise and flow analysis. In Sec. 3 the sta-tistical signatures of traffic jamming are discussed

numerically for both network types. In addition,we consider the transition to the congested traf-fic phase, where travel and waiting times tend todiverge, as a dynamical phase transition. Section 4discusses two optimization procedures, one withfixed network geometry, as above, and the otherwhich involves the network restructuring in order tominimize the packet travel times. We show that theemergent optimized structures in low and high den-sity traffic are statistically similar to the structuresdiscussed in Sec. 2. In Sec. 5 we give a summaryof the results and some open theoretical problemsposed by the numerical simulations and by empiri-cal measurements in real communication networks.

2. Stationary Traffic Flow

In the first part of this section we introduce twonetworks — a grown scale-free correlated network,which we call the Webgraph, and a homogeneousnetwork with a stretched-exponential degree distri-bution, which we call Statnet. We briefly summa-rize the structural properties of these networks thatare relevant to their transport processes. We thenintroduce the model of packet transport and per-form simulations on these networks with N = 1000nodes. The results we obtain for the statistical prop-erties of the transport in the free flow regime arepresented.

2.1. Structural properties ofthe networks

We consider two networks assembled by preferen-tial attachment [Dorogovtsev & Mendes, 2003] andin which rewiring [Tadic, 2001b] occurs.

The Webgraph is grown by sequentially addingnodes from i = 1, 2, . . . , N , with linking rulesthat involve preferential attachment and prefer-ential rewiring according to the time dependentprobabilities pin and pout. They are applied in thesubset of preexisting nodes at each growth step i.The linking probabilities depend on the currentnumber of incoming qin and outgoing qout links at anode [Tadic, 2001b]

pin(k, i) =α +

qin(k, i)M

(1 + α)i

pout(n, i) =α +

qout(n, i)M

(1 + α)i.

(1)


(a) (b)

Fig. 1. (a) The cyclic scale-free Webgraph, and (b) homogeneous Statnet.

Linking to a new node occurs with probability αand rewiring or adding a link from a previouslyexisting node occurs with probability 1 − α. Thesecompeting processes lead to an emergent struc-ture [Tadic, 2001b] which is different to the struc-ture obtained by common preferential attachment,[Albert & Barabasi, 2002; Dorogovtsev & Mendes,2003]. The parameters of the model α, α, and Mare responsible for the graph’s flexibility, connectiv-ity profile and clustering. In particular, when α = 1the graph is rigid (rewiring cannot occur) and scale-free, i.e. for M = 1 a scale-free tree and when M > 1a weakly clustered uncorrelated scale-free graphemerges as known in [Albert & Barabasi, 2002;Dorogovtsev & Mendes, 2003]. However, when α <1, these rules lead to power-law distributions of bothin-degree and out-degree [Tadic, 2001b], a largeclustering coefficient and link correlations [Tadic,2003], even for M = 1. When α = α = 1/4 thestructure is very similar to the world wide web[Tadic, 2001b]. This is why we call this network theWebgraph. An example of the emergent structureis shown in Fig. 1(a).

To grow the Statnet we apply the same ruleswith the probabilities in Eq. (1), however, with thefixed number of nodes i = N , among which linksare added sequentially. Multiple linking between thesame pair of nodes is not allowed. The Statnet emer-gent structure, with L = N = 1000, is also shownin Fig. 1(b).

A detailed quantitative analysis of the struc-ture reveals that both incoming and outgoinglinks behave the same statistically, and exhibit astretched-exponential degree distribution. In addi-tion, the clustering in this graph is small comparedto the Webgraph and link correlations are entirelyabsent. A comparison of the structural propertiesof both networks is given below.

Connectivity : The emergent connectivity of nodesin the evolving Webgraph can be obtained analyt-ically using linking probabilities in Eqs. (1). Bothfor in-coming and out-going connectivity we have apower-law profile according to [Tadic & Priezzhev,2002]

qκ(s,N) = Aκ

[(N

s

)γκ

− Bκ

]; (2)

after N nodes are added, s being the addition time.Here κ indicates “in”, “out”, or “total” links, forwhich different exponents are found [Tadic & Priez-zhev, 2002; Tadic, 2002]. For the purposes of thiswork, we consider that the transport along a linkin both directions is symmetrical. Therefore, thetotal node connectivity qtot ≡ qin + qout is relevant.In Fig. 2(a) we show the connectivity profile of allnodes in one Webgraph sample [shown in Fig. 1(a)].It exhibits a power-law decay with rank n ≡ n(i) ofa node i according to

X(n) ∼ n−γtot ; (3)


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Node Rank n(i)

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Cen

tral

ity B

(i)

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Con

nect

ivity

qto

t(i)

Fig. 2. (a) Node ranking of the total connectivity qtot = qin + qout, and (b) node ranking of the topological centrality B(i)in the Webgraph (�) and Statnet (©) structures. The fits are explained in the text.

with γtot = 0.867. This indicates a power-law con-nectivity distribution with the exponent τtot =1/γtot + 1 = 2.153. Similarly, node ranking accord-ing to the total connectivity in the Statnet, alsoshown in Fig. 2(a), is well fitted by a stretched-exponential curve

X(n) = An−τ exp[−

(n

n0

)σ]; (4)

with τ ≈ 0.22, σ ≈ 0.78, and n0 ≈ 200. The emer-gent probability distribution also obeys this law,Eq. (4).

Centrality : Further quantitative differences be-tween these two networks are found in the noderanking using the topological centrality measure[Newman, 2001; Zhou et al., 2005; Latora & Mar-chiori, 2004]. In Fig. 2(b) the ranked profile of topo-logical betweenness of nodes are shown. For theWebgraph the profile is a power-law Eq. (3) withthe exponent γB = 1, indicating the distributionof node betweenness as P (B) ∼ B−2. In the caseof Statnet the profile is again closer to a stretched-exponential form Eq. (4). In Fig. 2(b) one can alsoidentify the nodes belonging to the giant cluster,whose size is 403 nodes in the Webgraph, and 571in the Statnet (jump).

Clustering : Due to the rewiring, some nodes and/orsmaller clusters remain disconnected from the giantcomponent, whereas other nodes gain large connec-tivity and clustering. When the average numberof links per node is M ≡ L/N = 1 the averageclustering coefficient in the Webgraph appears tobe cc = 0.3601. In contrast Statnet, with sameaverage connectivity M = 1, has a much loweraverage clustering cc = 0.0075, which increaseswith M . The clustering profile is inhomogeneousin both networks. It obeys a power-law, shown inFig. 3(bottom) with an exponent close to γ =0.85 in the Webgraph, where most of the elemen-tary triangles are attached to the main hubs. Inthe case of Statnet a stretched-exponential profileis found.

Correlations : Another interesting feature whichmight influence the transport processes with searchis found in the link correlations of the Webgraph,which are entirely absent in Statnet. The com-puted correlation property measured by the aver-age out-degree of nodes attached to a node of givenin-degree, is shown in Fig. 3(top). The power-lawdecay with the exponent γc = 0.42 in the Web-graph indicates disassortative [Newman, 2003] linkcorrelations.


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Node Rank n(i)

10−1

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∆(n)

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qin

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<q ou

t>nn

Fig. 3. Link-correlations (top) and ranked clustering profile (bottom) for the Webgraph (�) with L/N = 1, and Statnet (©)with L/N = 20 corresponding to the same average clustering coefficient, cc = 0.36.

2.2. Implementation ofconstant-density traffic

We model the traffic of information packets on anetwork as a guided random walk between specifiedpairs of nodes [Tadic & Thurner, 2004; Tadic et al.,2004; Tadic, 2003] — the origin and destination(delivery address) of a packet. Once created, pack-ets navigate through the network using a local nnn-search rule [Tadic & Thurner, 2004, 2005; Tadic2003], in which the next-nearest-neighborhood issearched for the destination node. If a node findsthat a packet’s destination node is in its near-est neighborhood, it moves the packet directly toits destination. If the destination node is in thenext-nearest-neighborhood, but not the nearest-neighborhood, the packet moves in the correctdirection. If the destination node is not found inthe next-nearest-neighborhood, the packet moves toa randomly chosen neighboring node. Packets areremoved when they arrive at their destinations. Inthis section we model traffic with a fixed number ofmoving packets, which is set at the start of the sim-ulation, so that Np(t) = ρ, fixed for all t. Packetsthat arrive at their destinations and are removedfrom the network are replaced in the next time stepby the creation of the same number of new packets

at randomly chosen nodes. Of course, these pack-ets are given new destinations. When ρ = 1, thiscorresponds to the sequential random walk prob-lem. At density ρ > 1, packets interact by formingqueues at nodes along their paths. We assume finitemaximum queue lengths of H = 1000 at all nodes,and we employ a LIFO (last-in-first-out) queuingdiscipline.

Constant-density traffic is interesting becausethe limit of noninteracting packets ρ = 1 is math-ematically correctly implemented. This enablesquantitative analysis of the structure–dynamicsinterdependences without additional dynamiceffects [Tadic & Thurner, 2005]. Also, at finitetraffic density ρ > 1, the system is driven self-consistently, without external forcing. This situa-tion is suitable for the analysis of time-series andnoise fluctuations (see later), which are sensitiveto driving modes. Other driving conditions, e.g.constant posting rate R [Tadic & Thurner, 2004;Tadic et al., 2004; Arenas et al., 2001; Arenas et al.,2003], will be considered in Sec. 3 in connectionwith traffic jamming. When the traffic is drivenby creating at each time step a number of packetswhich is larger than network’s output rate, the net-work experiences congestion [Guimera et al., 2002a].


Driving at constant rate R is appropriate for aquantitative study of the congestion problem (seeSec. 3).

For the numerical implementation of the trans-port, we first generate the network and its adjacencymatrix is stored and remains fixed throughout thetransport process. If the graph is disconnected, asis the case with both Webgraph and Statnet, wetake care that the creation and destination nodesare within the same cluster. We study networks con-sisting of N = 1000 nodes. Starting with ρ = 100packets, which are created at random positions, thenetwork is updated in parallel. At each time step anode with a packet on it tries to move the top packetin its queue towards that packet’s destination node.A packet is moved to one of node’s neighbors, andjoins the top of its queue. If that neighbor is thepacket’s destination node, the packet is consideredas delivered and disappears from the network. Dur-ing the transport process, for each packet we keeptrack of its destination, current position and posi-tion in the current queue. In addition, for a subsetof labeled, NM = 2000, packets we keep track ofthe time that they spend in each queue before theyarrive at their destination. We compute the statisti-cal properties of the transport from the data gath-ered from the labeled packets once all of them havearrived at their destinations.

2.3. Global transport characteristics

The statistical properties of traffic depend on boththe network structure and traffic conditions. For afixed navigation protocol (nnn-search in this case),the efficiency of transport depends on the structuralcharacteristics of the underlying network and on theoverall traffic density. In particular, the travel timeof packets is determined by the length of the pathselected between the origin and destination node,and the waiting times at nodes along that path.That is, for a packet traveling along a path of lengthk, the travel time is given by

Tk =k∑

i=1

tw(i), (5)

where tw(i) is a packet’s waiting time on node ion that path. It should be stressed that both pathlengths and set of waiting times {tw} which con-tribute to the travel time T are outcomes of astochastic process. Then the functional central-limittheorem [Whitt, 2001] applies to the distribution oftravel times P (T ), computed along all paths on the

network [Tadic & Thurner, 2004]. The fact that onstructured networks the waiting times along a par-ticular path depend on the identity of nodes alongthat path adds to the complexity of the problem[Tadic & Thurner, 2004].

In the limit of noninteracting packets ρ = 1,the waiting times are tw = 1 for all nodes, thusthe entire travel time is determined by the geom-etry. The suitability of the navigation algorithmfor a given topology can then be measured by thedeviation of the actual path of the packet from theshortest path between the pair of nodes. In Fig. 4we show the distributions of the travel times forthe Webgraph and Statnet geometries and ρ = 100packets. For nnn-search these distributions exhibitpower-law tails with different exponents on the twonetworks. However, the largest difference appearsfor short travel times, where packets follow moreclosely the topological shortest path on the under-lying network. In this respect the node connectivityand centrality play an important role (see Fig. 2).The deviations from the shortest paths are experi-enced by packets which, i.e. local search being inef-fective, appear to perform a random walk in partsof the graph far away from their destinations. Thedistributions of travel times at large density ρ = 100appear to be well fitted with a power-law

P (T ) = AT−τT ; (6)

with τT ≈ 1.5 and an exponential cut-off (due to thefinite network size) for packets on the Webgraph,and a q-exponential distribution

P (T ) = Bq

[1 + (1 − q)

T

T0

]1/(1−q)

; (7)

with q = 1.47, for the case of transport on the Stat-net (cf. Fig. 4).

In processes in which diffusion dominates, thetopology of actual paths gives another view ofthe process. In particular, on structured networks,apart from the inhomogeneous connectivity, thenumber of short and long cycles may effect the dif-fusion of packets. The network profile of the shortcycles (triangles) in the Webgraph and Statnet areshown in Fig. 3. The packets that perform a ran-dom diffusion, i.e. when the local search is ineffec-tive for them, may return to a node, which hadalready been visited in the past. The distributionof the return-time intervals, ∆ti, for each nodei = 1, 2, . . . , N on the network, is given in Fig. 5 forboth network types. Broad distributions with char-acteristic power-law tails, suggest correlated events.


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T

10−8

10−6

10−4

10−2

100

P(T

)

10−8

10−6

10−4

10−2

100

P(T

)

Webgraph

Statnet

Fig. 4. Distribution of travel times in the Webgraph and the Statnet for a density ρ = 100.

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∆t

10−5

10−3

10−1

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103

P(∆

t)

Webgraph

Statnet

10−5

10−3

10−1

101

103

P(∆

t)

Fig. 5. Distribution of return times of packets in Webgraph and Statnet for a density ρ = 1.

Again the main difference between the two networktypes appears in the small return times of pack-ets. A power-law with small slope τ∆ = 0.66 for∆t � 800 can be related with an uneven popula-tion of short cycles and the active role of individ-ual nodes in the case of Webgraph. In the case ofStatnet the distribution can be fitted to the generalexpression in Eq. (7), with q = 1.26. On the Web-graph, the probability of long return times falls off

as P (∆t) ∼ (∆t)−3.26. When the packet density isfinite, the return time of node activity (where gener-ally a different packet is involved) is different on thetwo networks. The distribution is shifted towardsshorter values in the case of Webgraph. On Statnetthe form of the distribution changes.

For large packet density ρ � 1, the motionof a packet will be affected by other packets mov-ing through the same node, as mentioned above.


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Q

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P(Q

)

Webgraph

Statnet

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107

P[Q

]

Fig. 6. Distribution of queue-lengths in Webgraph and Statnet for a packet density ρ = 100.

Queues of packets then occur and a queuing dis-cipline sets the order of processing (LIFO, in thisparticular case). Apart from the traffic density, theneighborhood of the node determines the lengthof the queue at that node. In particular, on inho-mogeneous networks, hubs appear to receive morepackets compared to other nodes, due to their largeconnectivity. Since in the algorithm one packet isprocessed per time step, other packets remain inthe queue to be processed later (when no newpacket has been received). The distribution of queuelengths therefore reflects the network structure in aparticular way. A snapshot of queue-lengths Q, fortraffic density ρ = 100 in the two network structuresleads to the distributions shown in Fig. 6.

In the homogeneous Statnet most of the nodesare processing a similar number of packets, whichleads to a flat distribution of queues and a cut-offindicating that queues longer than Q ≈ 40 rarelyoccur. On the other hand, a large queue of Q = 80to 90, packets can be found on the hubs on the inho-mogeneous Webgraph with high probability. On therest of the nodes the queues are distributed witha power-law distribution, apart from very smallqueues at boundary nodes.

The queuing times of packets extend theirtravel times, thus reducing the overall traffic

efficiency [Tadic et al., 2004]. Note, that in the cur-rent implementation, with a constant number ofmoving packets, jamming cannot occur as long asthe traffic density ρ < H, where H is the maximumallowed queue length. However, due to long waitingtimes in queues, the travel times of packets given byEq. (5), can become very long, given by a power-lawdistribution, as shown in Fig. 4 (see also [Tadic &Thurner, 2004, 2005] for different types of graphsand search algorithms). The distributions of wait-ing times of packets on two network geometries aregiven in Fig. 7, for the traffic density ρ = 100. Theycan be described as power-law distributions,

P (tw) ∼ At−τww ; (8)

with different exponents, closely related to the tailsof the travel-time distributions in Fig. 4. Specifi-cally, in the case of Statnet the numerical valueof the exponent is τw ≥ 2, suggesting finite aver-age waiting times, and thus finite travel times forpackets on this network structure. In contrast, theaverage waiting time on the Webgraph for this traf-fic density is not bounded in a mathematical sense,with τw < 2. This implies a systematic increase ofthe travel times of packets on a large network withthis structure (with large measurement time). Thedistribution of waiting times and travel times of


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tw

10−8

10−6

10−4

10−2

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102

P(t

w)

10−8

10−6

10−4

10−2

100

102

P(t

w)

Webgraph

Statnet

Fig. 7. Distribution of waiting times in the Webgraph and the Statnet for a density ρ = 100.

individual packets provide a quantitative measurethat supports the conclusion that the Webgraphclass of networks, although much more efficientcompared to other scale-free network types [Tadic& Thurner, 2004; Tadic et al., 2004], is the less effi-cient of the two networks at high constant density ofpackets. More homogeneous structures, such as theStatnet, appear to perform better under these traf-fic conditions. It should be stressed that the waitingtimes as well as travel times are traffic propertiesmeasured for individual packets, which will dependon the type of queuing discipline used.

2.4. Noise and flow on networks

The observed queue lengths are compatible with thetemporal properties of node activity on the two net-works, shown in Fig. 8. While queues at importantnodes in the inhomogeneous Webgraph are long, thenumber of nodes that are simultaneously active issmall, fluctuating about an average value na ≈ 8.Compared to the more homogeneous Statnet for thesame traffic density, on average na ≈ 80 nodes areprocessing a packet simultaneously, leading to shortqueues at all nodes.

Further quantitative analysis of the time-seriesreveals the differences in the packet processingof the two classes of networks. In particular,

long-range correlations (anti-persistence) in thenumber of active nodes develops on both networks.However, in the conditions of constant packet den-sity the fluctuations on the Statnet appear to bemore correlated compared to the Webgraph. Thepower spectrum exhibits an asymptotic power-lawbehavior

S(f) ∼ f−φ, (9)

shown in Fig. 8 (top panel), with φ = 1.1 for Stat-net and φ = 1.4 for the Webgraph. Therefore, anincreased traffic density leads to stronger correla-tions in node activity on the more homogeneousStatnet [Tadic & Thurner, 2006].

The observed differences that individual nodeson each network play in the transport processes arefurther quantified by analysis of the noise fluctua-tions. We measure the number of packets hi(t) thata node i processes within a fixed time window ofTWIN = 1000 time steps. We consider simultane-ously (multichannel noise analysis) a set of time-series {hi(t)} for all i = 1, 2, . . . , N nodes andt = 1, 2, . . . , 500 successive time windows. We deter-mine the standard deviation σ(i) for each of Ntime-series (i.e. for each node) and plot it againstthe time-averaged value 〈h(i)〉 ≡ 〈hi(t)〉t, where theaverage is taken over all time windows for each node


0 1000 2000 3000 4000 5000time

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# ac

tive

node

s

Statnet

Webgraph

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f

10−6

10−5

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S(f)

WebgraphS=78f

−1.44

StatnetS=18f

−1.1

Fig. 8. Temporal fluctuations of the number of active nodes in Webgraph and Statnet (lower panel) and their power spectra(top panel) for a density of ρ = 100 packets (from [Tadic & Thurner, 2006]).

separately. The general scaling relation [Argollo deMenezes & Barabasi, 2004]

σ(i) ∼ 〈h(i)〉µ, (10)

holds for all nodes in the network for our constant-density traffic. However, we find that the scalingexponents may depend on the traffic density in theinhomogeneous Webgraph. The results are shownin Fig. 9 for the two network structures and threedifferent traffic densities ρ. In particular, when thepacket density is high, the number of packets pro-cessed by the hub nodes increases, resulting in theincreased fluctuations at these nodes. When densityis comparable with the maximum buffer size (in ourcase ρ = H = 1000), temporary congestion mayoccur at the main hubs and at other nodes of largeconnectivity, whereas the rest of the network func-tions in the free flow regime. In this situation, thenoise fluctuations at more important nodes appearto be in the µ = 1 class, as opposed to the rest of thenetwork, where the fluctuations follow the law withµ = 1/2, as shown in Fig. 9. Due to the absence ofhubs in the Statnet the fluctuations remain homo-geneous and in the µ = 1/2 class for all packetdensities.

Our results demonstrate that inhomogeneousnoise fluctuations may occur in the self-regulated

(constant density) traffic away from the two uni-versality classes defined in [Argollo de Menezes &Barabasi, 2004]. (See also [Valverde & Sole, 2004]for traffic self-regulation with a local feedback.) Inour study the explanation of these nonuniversalnoise fluctuations is found in how the network struc-ture handles high density traffic. These findings arein agreement with a recent detailed study of theempirical data of the Internet traffic and in a specif-ically designed model in [Duch & Arenas, 2006],which revealed that when the traffic density is high“enough”, then other parameters of the dynamicsand structure play a role in determining the noiseproperties. In particular, the exponent values mayvary between 0.5 and 1, depending critically onthe width of the time window and on the queu-ing times of packets [Duch & Arenas, 2006]. (Sim-ilar time-window effects are discovered in [Eisler &Kertesz, 2006] in the stock market dynamics). Com-pared with earlier models which tackle the scaling oftraffic noise in the Internet [Sole & Valverde, 2001;Argollo de Menezes & Barabasi, 2004; Valverde &Sole, 2004; Duch & Arenas, 2006], the traffic modelthat we study here is more complex in that pack-ets are navigated to specified destinations, and thatboth travel times and queuing times of packetsare self-consistently determined and appear to have


103

102

101

100

101

102

103

104

<h(i)>

10−2

10−1

100

101

102

σ(i)

ρ=1ρ=100ρ=1000µ = 1/2

Webgraph

Statnet

10−2

10−1

100

101

102

103

σ(i)

ρ=1ρ=100ρ=1000µ = 1/2µ = 1

Fig. 9. Universal noise fluctuations for traffic with constant-density ρ = 1, 100, and 1000 packets and with nnn-search on theWebgraph and Statnet. Lines indicate scaling dependence in Eq. (10), with slopes µ = 1/2 and µ = 1.

power-law distributions (cf. Figs. 4 and 7). In thefuture, we would hope to determine how these trafficproperties contribute to the observed nonuniversal-ity in noise fluctuations.

3. Traffic Jamming and Structure

A crucial feature of transport networks is thatthey cease to function efficiently when jamming ofnodes occurs. Jamming is characterized by a drasticdecrease of efficiency, in particular a nonstationaryincrease of the load as a function of R. Jammingoccurs as a transition at a critical rate Rc. It is of rel-evance to find alternative indicators or signatures,which are able to signal the occurrence of jamming.In particular we focus on exploring the possibilityto predict the “distance” to the jamming transitionfrom the activity time-series of individual nodes.

As will be discussed in Sec. 4, networks of dif-ferent structures perform differently when the traf-fic density is varied. In the constant-density trafficρ = 100 we show here that the network output,defined as the number of packets delivered per timestep nd(t), is larger in the homogeneous Statnetcompared with the scale-free Webgraph. In Fig. 10we display the time variation of the network outputnd(t) rate for the two networks. The average outputrate λ is defined as

λ ≡⟨

dnd(t)dt

⟩. (11)

In the Webgraph λ = 0.37 we find packets per timestep, compared with Statnet, where λ = 0.71. Thisdifference suggests that with an imposed traffic den-sity and a constant number of moving packets ofρ = 100, the traffic on the Webgraph topology iscloser to jamming than the Statnet topology. In thissection we will explore in more detail the statisticalsignature of traffic near jamming.

3.1. Queuing and jamming atdifferent topologies

The approach to the jamming regime is most appro-priately studied with a constant posting rate R.Then the temporal variations in the network loadNp(t) are given by

Np(t) = Rt − nd(t). (12)

When the posting rate R increases so that it exceedsthe average output rate, R > λ, the excess packetsaccumulate on the network leading to a systematicincrease in the network load Np(t), although thenetwork continues to deliver packets in some way.The average accumulation rate is then obtainedfrom Eq. (12) as

J ≡⟨

dNp(t)dt

⟩= R − λ, (13)

which is reminiscent of the congestion conditionJ/R = 1 − λ/R, often found in queuing theory


0 5000 10000 15000Time t

0

5

Del

iver

y ra

te

Webgraph

Statnet0

5

Del

iver

y ra

te

Fig. 10. Number of packets delivered per time step nd(t) against time t for fixed density of moving packets ρ = 100 inWebgraph and Statnet. Dotted lines indicate values of average output rate λ.

for a single-server queue [Allen, 1990; Whitt, 2001].It should be stressed, however, that here the loadNp(t) applies to the whole network, which con-sists of many interacting queues. The excess loadincreases the queues at different nodes, dependingon each node’s topological centrality and its impor-tance in the particular transport process. There-fore, subtle interactions between different queuesand their dynamical correlations contribute to thejamming process, which are different for differentnetwork topologies. This aspect of network conges-tion will be discussed in detail later. We first demon-strate how the statistical properties of the trafficchange with increased posting rate by simulatingtraffic on the Webgraph.

In Fig. 11(a) we show the time-series of thenetwork load Np(t) for three representative post-ing rates R =0.25, 0.35 and 0.45 on the Webgraph.Compared to the time-series of the number of activenodes in Fig. 8, here we take into account thateach active node has a queue of packets of a lengthQi(t) ≥ 1, which contributes to the overall networkload, Np(t) =

∑Ni Qi(t). Two of the time-series

in Fig. 11(a) are stationary, however, the averageload is increasing with the posting rate R. Anotherqualitative difference is that, for the larger postingrate, R = 0.35 in this case, a temporary jammingmay occur that lasts for approximately 60000 timesteps, and eventually resolved by the system itself.This “crisis”-like behavior is one of the manifesta-tions of the approach to a congested flow regimeat a critical posting rate Rc. A systematic analysis

of the traffic on Statnet for different posting ratesR reveals that the average load 〈Np(t)〉 is muchlarger, compared to the Webgraph, but also thatthe critical rate where the jamming occurs is twicethe critical rate of the Webgraph (see below). InFig. 12 network-load time-series are shown for tworepresentative posting rates just below the respec-tive jamming point in both networks.

As shown in [Tadic et al., 2004], on approach-ing the jamming point, numerous manifestations ofdeveloping congestion can be measured statistically.In particular, increased waiting times of packets andcharacteristic changes in the distribution of waitingtimes and network loads are detected, as well as inthe correlations of the activity and load time-series.For the purpose of this work, we discuss only thesystematic changes in the network-load time-series.The power-spectra of the network-load time-seriesfor the traffic on Webgraph are shown in Fig. 11(b)for different values of the posting rate in the rangeR = 0.005 to R = 0.4. The remarkable feature ofthe spectra is the systematic decrease in the correla-tions (antipersistency), measured by the increasedscaling exponent φ in Eq. (9) from φ = 1.18 at low-est considered posting rate, to φ = 2, at the jam-ming threshold. This gives the numerical evidencethat the onset of congested state is characterized bya loss of long-range correlations in packet streams.This feature seems to apply generally. The transi-tion to the congested state and the properties of thetraffic in the congested regime, however, depend onthe topological details of the network.


0 20000 40000 60000 80000 1e+05 Time t

0

100

200

300

400

Net

wor

k Lo

ad N

p(t)

(a)

100

101

102

103

104

105

Frequency f

10−6

10−5

10−4

10−3

10−2

10−1

100

101

102

103

104

Pow

er S

pect

rum

S(f

)

(b)

Fig. 11. (a) Network-load Np(t) time-series for transport on the Webgraph with nnn-search and three posting rates indicating(bottom to top) free flow, flow with “crisis”, and occurrence of jamming. (b) Power-spectra of Np(t) time-series for severalposting rates below jamming threshold. Dashed and dotted lines indicate limiting slopes with φ = 1 and φ = 2, respectively.The jamming transition corresponds to φ = 2.

In order to avoid possible confusion regardingthe role of the transition point to the congestedstate, here we would like to stress that the network-load time series in our model exhibits long-range

0 50000 1e+05Time t

0

50

100

150

200

Net

wor

k Lo

ad N

p(t)

Webgraph

Statnet

Fig. 12. Network-load time-series close to jamming transi-tions in Webgraph and Statnet.

correlations (with an exponent φ < 2) in a widerange of posting rates below the jamming transition.As we show in the next subsection, these correla-tions are strictly related to the structural complex-ity of the network.

3.2. Transition to congestedtraffic state

Technically, the transition point can be identifiedwith φ(Rc) → 2 for a given network geometry.A characteristic slow-down of the dynamics at acongested node, typically a hub in the Webgraph,occurs in that only one packet can move-in fromone of the neighbor nodes, and a large number ofpackets at other neighboring nodes are waiting untilone packet moves out of the hub in the next timestep. Study of long time-series for different postingrates reveals how the decorrelation occurs on differ-ent network structures. In Fig. 13 we show values ofthe exponent φ against R in the case of Webgraphand Statnet. It shows first that large variations in


0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Posting Rate R

0.9

1

1.1

1.2

1.3

1.4

1.5

1.6

1.7

1.8

1.9

2

2.1

Exp

onen

t φ;

<dN

p(t)

/dt>

+1

φ

<dNp(t)/dt> +1

Fig. 13. (Upper curves) Correlation exponent φ defined inEq. (9) and (lower curves) jamming rate 〈dNp(t)/dt〉 (shiftedby 1) for different values of posting rates R in traffic on theWebgraph (filled symbols) and Statnet (open symbols).

the strength of correlation occur in load time-serieson the Webgraph, in contrast to the time-series onthe Statnet, which exhibit weaker but stable cor-relations over a wider range of posting rates. Thevalues of the critical rates are Rc ≈ 0.4 for trafficon the Webgraph, whereas Rc ≈ 0.8 for the Statnet,that roughly coincide with the time-series decorre-lation rates.

Additional quantitative characteristics of thetransition to the congested state can be obtainedby measuring the average jamming rate, J , whichis defined in Eq. (13). In Fig. 13 we show system-atic dependences of the jamming rate J from theexternally imposed posting rate R for traffic onboth types of networks. It reveals that the criti-cal posting rates Rc = 0.4375 ± 0.0125 for trafficon the Webgraph, and Rc = 0.75 ± 0.0125 in thecase of Statnet, at which the jamming transitionstarts to appear are closely associated with a com-plete decorrelation in the respective time-series (φreaches the value of 2 in both cases). Apart fromthe differences in values of the critical rates Rc,the onset of jamming seems to occur abruptly inboth networks. Additional differences due to net-work topology are seen in the character of thejammed traffic. Namely, the slopes of the curvesfor J(R) above the jamming point in each case aredifferent. This suggests that in the congested statethe delivery rate λ, according to Eq. (13) drops toλ/R = 1 − J/R, where J/R = 0.7 for the case ofWebgraph, and J/R = 0.5 for the case of Statnet,

are slopes of the curves J(R) in Fig. 13. Therefore,in the congested state the Webgraph continues toprocess about 30% of posted packets, whereas theStatnet manages to deliver about 50%.

At this point, it is interesting to compare theconstant density traffic, ρ = 100 studied in pre-vious section, with the picture of jamming on thetwo topologies. In the constant density traffic aneffective posting rate 〈R〉 emerges, which keeps thebalance of the network’s output rate. According toFig. 10, for ρ = 100 the effective posting rates,〈R〉 = λ = 0.37 for the Webgraph, and 〈R〉 = λ =0.71 in the case of Statnet, are below the respectivejamming point in both topologies.

4. Optimized Transport

Perhaps the most important question in transportoptimization is to identify the optimal performanceof a network, once certain constraints such as searchmode and queue protocols are specified. Here wereview two ideas that address this question fromdifferent standpoints. The first develops a methodto identify optimal paths in different networks byusing maximum flow trees [Tadic & Thurner, 2006],the other actually optimizes network structure fora given protocol and search depth [Guimera et al.,2002b].

4.1. Optimal paths on fixedtopologies

By simulating a large number of packets we recordthe number of walks along each link (dynamicflow) and through each node (dynamic noise) inthe network. Obviously, the inhomogeneity of nodeswith respect to their local network environment, asshown in Fig. 2, makes the flow and noise fluctua-tions different on each network structure. To obtaina quantitative analysis of flow on the network links,here we construct a maximum-flow spanning tree,on which each node is connected to the rest of thenetwork nodes via its maximum-flow link. Imple-menting a greedy algorithm, we determine the treesfrom maximum-flow in the ρ = 1 limit on the twonetworks. The trees are shown in Fig. 14.

The structure of these trees reflects both theunderlying network geometry and how that geom-etry affects transport with given navigation rules–local nnn-search. In the case of the Webgraph thetree exhibits a scale-free topology, suggesting a cer-tain degree of compatibility between the traffic and


(a) (b)

Fig. 14. Dynamic maximum-flow spanning tree for traffic density ρ = 1 on the (a) cyclic scale-free Webgraph, and (b)homogeneous Statnet.

the structure. Similarly, for the Statnet the treeshows some degree of inhomogeneity that corre-sponds to the weaker inhomogeneity of the under-lying graph.

The maximum-flow spanning trees representthe union of maximum-flow paths on the underlyingnetwork structure. In Fig. 15 we show distributionof the lengths of all such paths on the two trees thatare shown in Fig. 14. Once again, differences in thegraph topologies and thus in their maximum traf-fic trees manifest themselves in the statistics of themaximum-flow paths. For instance, the average dis-tance along such paths on the Webgraph and Stat-net differs by a factor of about 5, the maximumdistance differs by a factor of about 3.

The message is that for a fixed navigation pro-tocol the underlying network structure determinesthe topology of optimal paths for packet trans-port. These paths consist of links that appearedto be locally optimal choices for packets, withoutany global feedback. The quantitative differences inthese topologies can be determined in the limit ofnoninteracting packets (as shown in Fig. 14). How-ever, how these optimal paths are used by packetswhen the traffic density is increased, again dependson the properties of the network queues. In partic-ular, the traffic efficiency due to very short pathsbetween pairs of nodes, as on scale-free topologies,can be hindered by the occurrence of large queuesat hubs, discussed in Sec. 3. In the following we

0 10 20 30 40Path length

100

101

102

103

104

105

Fre

quen

cy

WG SN

Fig. 15. Distance distributions on the maximum-flow span-ning trees for the Webgraph and the Statnet for ρ = 1, andnnn-search [Tadic & Thurner, 2006].

show what network topologies emerge as globallyoptimal structures in relation to varying traffic den-sity [Guimera et al., 2002b].

4.2. Traffic optimization bynetwork restructuring

In Secs. 2 and 3 we have demonstrated that trans-port efficiency on a given network topology maycrucially depend on the traffic density. Specifically


in strongly inhomogeneous networks, such as theWebgraph, the structural characteristics which areadvantageous at low traffic density — occurrenceof powerful central nodes, may appear as weak-nesses at high density traffic. In this subsection wewould like to draw attention to a more systematicway to obtain an optimal structure for a given den-sity using network reconstruction. The suitable for-malism was introduced in [Guimera et al., 2002b],where global structure optimization is performedstarting from data about individual packet flow. Inthis respect, the formalism simultaneously accountsfor both search and congestion aspects of the infor-mation flow on networks. The basic arguments arepresented below. For more details see the originalreference [Guimera et al., 2002b].

As before the focus is on a single informationpacket at node i whose destination is node k. Theprobability for the packet to go from i to a new

node j in its next movement is pkij. In particular,

pkkj = 0, ∀j so that the packet is removed as soon

as it arrives at its destination. The probability pkij

depends on the network topology, given by the ele-ments of the adjacency matrix cij , and on the searchalgorithm. The three cases with local search modifythe transition probability pk

ij in the following way:For a random walk (searched depth zero) we have

pkij =

cij∑j

cij

, (14)

and for a random walk with nearest neighbor search

pkij = cikδjk + (1 − cik)

cij∑j

cij

. (15)

Finally, for the the nnn-search used in Secs. 2 and3, it corresponds to

pkij =

δjk if cik = 1cijcjk∑

j

cijcjk

if cik = 0 and∑

j

cijcjk > 0

cij∑j

cij

if cik = 0 and∑

j

cijcjk = 0

(16)

Following [Guimera et al., 2002b], one can thendetermine the dynamic betweenness of nodes and,subsequently, the network load Np(t), under certainfairly general conditions. When the search is Marko-vian, i.e. pk

ij does not depend on previous positionsof the packet, the probability of going from i to jin n steps is given by

P kij(n) =

∑l1,l2,...,ln−1

pkil1p

kl1l2 . . . pk

ln−1j . (17)

This definition allows calculation of the averagenumber of times, bk

ij , that a packet generated ati and with destination at k passes through j. Thiscan be expressed in matrix notation as

bk =∞∑

n=1

P k(n) =∞∑

n=1

(pk)n = (I − pk)−1pk, (18)

where the matrices bk, P k(n) and pk have elementsbkij, P k

ij(n) and pkij respectively and I is the iden-

tity matrix. The effective (dynamic) betweenness,Bj, of node j is then defined as the sum over all

possible origins and destinations of the packets onthe graph,

Bj =∑i,k

bkij . (19)

When the search algorithm is able to find theminimum paths between the origin and destina-tion nodes, the effective betweenness will coincidewith the topological betweenness [Newman, 2001;Zhou et al., 2005; Latora & Marchiori, 2004], con-sidered in Sec. 2.

If packets are generated at random and inde-pendently with a probability R, or at each nodewith probability r = R/N , and if the queuing dis-cipline is given by a random queue M/M/1, ratherthan the more complicated LIFO used in Sec. 2, itcan be shown that the time averaged load of thenetwork is [Allen, 1990; Guimera et al., 2002b]

Np =N∑

j=1

rBj

N − 1

1 − rBj

N − 1

. (20)


(a) (b)

Fig. 16. Optimal networks for (a) low-density traffic, and (b) high-density traffic. (Data: courtesy A. Dıaz-Guilera.)

This solution has two interesting limiting cases: Forsmall values of r the average load is proportionalto the average effective distance [Guimera et al.,2002b]. On the other hand, when r approaches acritical rate rc, most of the load of the networkcomes from the most congested node, and therefore

Np ≈ 1

1 − rB∗

N − 1

; r → rc, (21)

where B∗ is the effective betweenness of the mostcentral node, and assuming that the jump proba-bilities pk

ij do not depend on the congestion state ofthe network.

Equation (20) relates a dynamical variable, theload, with topological properties of the networkand of the search algorithm. Hence, the dynam-ical optimization procedure of finding the struc-ture that gives the minimum load is reduced toa topological optimization procedure where thenetwork structure is characterized by its effective(dynamic) betweenness distribution. In [Guimeraet al., 2002b] the problem of finding optimal struc-tures was considered for a purely local search,Eq. (14), using a generalized simulated annealingprocedure, as described in [Tsallis & Stariolo, 1994;Penna, 1995]. Surprisingly, it was found that thereare only two types of structures that can be opti-mal for a local search process: star-like networks forposting rates below a characteristic rate R∗, r < r∗

and homogeneous networks for r > r∗. The net-works are shown in Fig. 16.

The network structures which result from theoptimization process for two limits of the trafficdensity, Fig. 16, exhibit a remarkable similarity withgraphs considered in Sec. 2, in particular, with theircore-graphs (graphs without single-link nodes). Thiscomparison helps us to identify the main topologi-cal property of the graphs in Fig. 1 that is respon-sible for the transport efficiency. In particular, itsuggests that the large clustering in the Webgraphnear the main hubs is responsible for the increasedefficiency of transport at low traffic density [Tadic& Thurner, 2004; Tadic et al., 2004] (see also Sec. 3)compared to other scale-free networks. Similarly,the absence of clustering and a narrow distributionof the connectivity in the Statnet, makes it suitableto support a large-density traffic, in analogy to thehomogeneous network in Fig. 16.

5. Open Problems and Conclusions

5.1. Summary

We have shown using two complementaryapproaches that two different classes of networkstructure give rise to different traffic properties.The classes of network structures we consideredwere scale-free graphs with a significant hub nodeand high clustering (called Webgraph), and a muchmore homogeneous network with a quickly decaying


degree distribution (called Statnet). These classeswere identified in two ways. Firstly, by directlyimplementing a fairly realistic protocol for packettransport on these networks we demonstrated witha number of quantitative characteristics that thetraffic behaves differently on the two networks. Sec-ondly, making use of the results of network opti-mization in [Guimera et al., 2002b], we concludedthat these two network classes appear to be rep-resentative of optimal structures for low and hightraffic density, respectively.

A comparative study of packet transport onthese two networks revealed how traffic propertiesdepend on the network structure and on the packetdensity. We considered both free moving and con-gested traffic, and driving conditions with both aconstant packet density and with a constant post-ing rate. A summary of the traffic characteristicsidentified is given below:

Power-law tails : In the statistics of individualpackets the distributions of travel time, P (T ), andwaiting time, P (tw), exhibit power-law tails on bothnetwork types, however, with different slopes (cf.Figs. 4 and 7). Further differences appear at shorttravel times, where transport on the Webgraphmostly follows the shortest path between source anddestination nodes, while the times up to a charac-teristic time T0 ≈ 10 steps are all equally probableon Statnet. Consequently, the distribution P (T ) fitsa q-exponential [Tsallis, 1988] in the homogeneousStatnet and a true power-law in the scale-free Web-graph. The distributions of return-time, P (∆t), haspower-law tails with a large slope on both networks.The main differences between the networks appearat short return times (see Fig. 5), where the inho-mogeneous betweenness of nodes in the Webgraphplay an important role.

Universal noise fluctuations : The traffic noise aver-aged over a suitably chosen time window follows theuniversal law in Eq. (10) in both network classes.In the free flow regime we find µ = 1/2 law inboth network types, regardless of the differencesin their homogeneity and clustering characteristics.However, when the traffic jamming occurs at largedensity, the law changes towards µ = 1, for the busi-est hub nodes in the inhomogeneous scale-free Web-graph, whereas no qualitative changes are observedin the homogeneous Statnet.

Antipersistent time-series : The network load time-series Np(t) in both network types are found to beantipersistent in the free flow regime. The degree

of correlations, however, are considerably different(Fig. 13), depending on traffic density (or post-ing rate). The noise correlations on Webgraph varyfrom very strong, at low density, to weak corre-lations, at the jamming threshold Rc ≈ 0.4. Inthe case of homogeneous Statnet the noise corre-lations are generally weaker but stable in a widerange of posting rates until the jamming thresh-old, Rc ≈ 0.8, when only short-range correlationssurvive.

Jamming transition : A jammed traffic regime,characterized by unbalanced increase of networkload occurs on both networks at their critical post-ing rates Rc, mentioned above, Fig. 13. The appear-ance of the nonzero jamming rate λ Eq. (11) is quitesudden for finite sized networks of both classes. Fora given network type the value of the critical thresh-old Rc depends strongly on the efficiency of thesearch algorithm [Tadic & Thurner, 2004]. Specif-ically, the critical rate Rc ≈ 0.4 in traffic withnnn-search on the Webgraph is particularly highcompared to random diffusion on the same graph ortraffic on scale-free trees, where the critical jammingrate is of the order Rc ≈ 10−3 [Tadic & Thurner,2004].

Traffic in the congested state : The advantage of thehomogeneous structure at high traffic density con-tinues to be seen in the congested regime. However,the difference in the efficiency is only about 20%.This can again be attributed to the efficient nnn-search of the Webgraph geometry.

Optimal flow paths : The union of the optimal pathslearnt from the traffic history, summarized at themaximum-flow spanning trees of the two networktypes in Fig. 14 shows the advantage of the scale-free structure at low density traffic. Due to theefficient search, the maximum-flow tree retains thescale-free character and the statistics of the optimalpath lengths in Fig. 15 resemble those of the topo-logically shortest paths on the graph. On the otherhand, paths are much longer on the homogeneousStatnet, independent of the traffic density.

For a fixed network geometry further, althoughlimited, optimization may be achieved by improv-ing the search algorithms [Kujawski et al., 2006].Transport processes on other types of networks, cel-lular networks [Suvakov & Tadic, 2006], gradientnetworks [Toroczkai & Bassler, 2004; Park & Lai,2005], river networks [Banavar et al., 1999], or gen-erally trees and directed graphs, are subject of addi-tional constraints that may result in qualitatively


different behavior. These networks are not consid-ered in the current work.

5.2. A comment regarding“realistic” protocols

Among various transport networks the Internet isa particularly accessible and attractive example.In recent years measurements in both the Inter-net structure and the information packet TCP/IPtraffic dynamics on the Internet have been car-ried out and their inter-relations discussed [Csabai,1994; Takayasu et al., 1996; Takayasu et al., 2000;Willinger et al., 2002; Abe & Suzuki, 2003; Doyleet al., 2005]. These have either considered ping timestatistics, that is a packet’s round trip time fromsource to destination and back again, or the statis-tics of the load on a particular server, router orcable. Almost all these studies have found scalinglaws and long range correlations in a range of traf-fic time-series [Csabai, 1994; Takayasu et al., 1996;Takayasu et al., 2000]. Examination of the powerspectra has allowed the identification of two dif-ferent regimes with free flow or jammed traffic. Itwas shown [Willinger et al., 1997] that the power-law behavior of the distribution of packet inter-arrival times has a significant impact on packetqueue statistics, and consequently on the overalltraffic performance.

In addition, the scale-free nature of the Internetstructure [Vazquez et al., 2002; Doyle et al., 2005]has been fully characterized. The structural charac-teristics on the autonomous systems level, in par-ticular, the degree distribution, high clustering andlink-correlations, are statistically similar to thoseof the prototype network Webgraph discussed inSec. 2. Beyond the practical importance of researchinto the properties of the Internet, the underlyingcause of self-similarity and criticality in the Inter-net’s structure and information traffic is still a sub-ject of debate involving researchers in a broad rangeof scientific disciplines. Among the goals of thisresearch is to determine universality classes of thescale-free behavior and to unravel the mechanismsof the structure — dynamics interdependences.

Examining our results in Sec. 2, obtained withthe local nnn-navigation algorithm, we can con-clude that the statistical features of the traf-fic, in particular the power-law distributions, thedegree of correlations in the packet streams, andthe onset of jamming, are related to the actualInternet structure. Furthermore, our results suggest

that no substantial improvement in traffic efficiencywill be achieved by implementing a long-rangesearch beyond the “critical horizon” of the scale-free network, which is depth level two in our Web-graph (more detailed analysis is given in [Tadic &Thurner, 2005]). In the study by [Valverde & Sole,2004], who first introduced the term “critical pathhorizon” in this context, the sharp transition wasfound at the depth level four. The difference in thecritical horizons in these two scale-free networks canbe attributed to the better “searchability” of thecorrelated scale-free Webgraph, compared to uncor-related scale-free network used in [Valverde & Sole,2004].

Of crucial importance is the centrality of nodes,which pre-determines the size of queues, and hencethe traffic jamming and an increased risk of packetloss. The onset of traffic congestion seems to occursuddenly in the prototype network, and is proba-bly related to the size of the giant cluster. However,we observed a kind of traffic “crisis” behavior withlarge load fluctuations before the actual jammingstarts. The statistical indicators of the traffic behav-ior, determined in previous sections, systematicallychange with the increased network load (traffic den-sity). Hence, the following four phases can be clearlyidentified: free flow, crisis, jamming threshold andcongested traffic. Therefore, an effective controlmechanism, that would eventually lead to increasedtraffic efficiency and security, may be developedthrough a systematic monitoring of one or more ofthese indicators.

5.3. Some open theoretical problemsin transport on networks

Quantitative properties of transport on a networktopology depend in different ways on the networkgeometry and on search algorithm and type of queueson that geometry. The actual dependence on thesearch and queuing cannot be considered indepen-dently of the underlying network structure [Tadic& Thurner, 2005]. Another subtle factor whichdetermines the transport properties is the drivingmode. Specifically, large fluctuations of the postingrate may influence noise properties. More seriously,they may drive the network out of a stationary flow,where some statistical properties cannot be deter-mined mathematically correctly.

For the theoretical purposes we have imple-mented in Sec. 2 a noninvasive self-consistent driv-ing, which preserves the number ρ of moving packets


in time. The stationarity of time-series is thus guar-anteed, and jamming does not occur when ρ is nottoo big, as shown above. In this regime we can deter-mine numerically the correct statistical propertiesof the traffic. In the limit ρ → 1 the outcomes aredirectly attributed to the network structure. Theresults of numerical simulations are given in Sec. 2.However, clear theoretical concepts behind thesenumerical results still remain elusive, in particularrelated to the following questions:

• Universality classes of travel time distributions.• Power-laws in the waiting time distribution.• Queue interactions in the correlated network

environment.• Return-time distributions structural dependences

and occurrence of the q-exponential.• Robustness of the universal noise fluctuations,

related to traffic density and network structure.• Universality of the dynamic jamming transition.

In traffic models, as in many other dynami-cal systems, power-law behavior is often attributedto (dynamic) phase transitions, separating differ-ent attractors of the dynamics. A particular exam-ple of this is the jamming transition in the trafficmodel on a compact lattice studied by Sole andValverde [2001]. However, when the underlyingnetwork is sparse and strongly inhomogeneous,additional mechanisms of correlations may becomeactive, leading to the attractive states with self-similar dynamics away from the phase transition.In this paper, by comparing traffic on differentnetworks, we have found substantial evidence thatthe self-similar dynamic properties (at and away ofthe jamming transition) are shaped by the networkgeometry.

In conclusion, the dynamic characteristic oftransport processes on networks, appear to be, to alarge extent, predesigned by the underlying networkstructure, where certain structural characteristicsplay a dominant role. However, these structure–function interdependences are strongly determinedby the dynamic conditions, in particular, traffic den-sity. In this respect, two large classes of networkswith an optimal function are now identified —clustered scale-free networks, and homogeneous, orweakly structured, unclustered networks. With thelarge-scale numerical simulations in this work wehave shown that different quantitative character-istics of the traffic emerge in these two networkclasses. This leaves open the question about theactive principle which shapes the dynamics within

each of these network classes. We hope that ourinvestigations will initiate further theoretical andpractical research to address this question.

Acknowledgments

B. Tadic acknowledges support from the ProgramP1-0044 of the Ministry of Higher Education, Sci-ence and Technology of the Republic of Slovenia, STfor FWF Project P17621, Austria. Further, partialsupport from the COST Action P10 Physics ofRisk is acknowledged.

References

Abe, S. & Suzuki, N. [2003] “Omori’s law in the internettraffic,” Europhys. Lett. 61, 852–855.

Agrawal, H. [2002] “Extreme self-organization in net-works constructed from gene expression data,” Phys.Rev. Lett. 89, 268702.

Albert, R. & Barabasi, A.-L. [2002] “Statistical mechan-ics of complex networks,” Rev. Mod. Phys. 74, 47–111.

Allen, O. [1990] Probability, Statistics and QueueingTheory with Computer Science Application (Aca-demic Press, NY).

Arenas, A., Dıaz-Guilera, A. & Guimera, R. [2001]“Communication in networks with hierarchicalbranching,” Phys. Rev. Lett. 86, 3196–3199.

Arenas, A., Cabrales, A., Dıaz-Guilera, A., Guimera, R.& Vega-Redondo, F. [2003] “Search and congestion incomplex networks,” in Proc. Conf. Statistical Mechan-ics of Complex Networks, Lecture Notes in Physics,eds. Pastor-Satorras, R., Rubi, M. & Dıaz-Guilera,A., Vol. 625 (Springer Verlag, Berlin), pp. 175–194.

Argollo de Menezes, M. & Barabasi, A.-L. [2004] “Fluc-tuations in network dynamics,” Phys. Rev. Lett. 92,028701.

Ashton, D. J., Jarrett, T. C. & Johnson, N. F. [2005]“Effect of congestion costs on shortest paths throughcomplex networks,” Phys. Rev. Lett. 94, 058701.

Banavar, J., Maritan, A. & Rinaldo, A. [1999] “Sizeand form in efficient transportation networks,” Nature399, 130–132.

Boccaletti, S., Latora, V., Moreno, Y., Chavez, M. &Hwang, D.-U. [2006] “Complex networks: Structureand dynamics,” Phys. Rep. 424, 175–308.

Boguna, M., Pastor-Satorras, R. & Vespignani, A. [2003]“Epidemic spreading in complex networks with degreecorrelations,” in Statistical Mechanics of ComplexNetworks, Lecture Notes in Physics, eds. Pastor-Satorras, R., Rubi, M. & Dıaz-Guilera, A., Vol. 625(Springer Verlag, Berlin).

Boss, M., Elsinger, H., Summer, M. & Thurner, S. [2004]“The network topology of the interbank market,”Quant. Fin. 4, 677–684.


Burda, Z., Jurkiewicz, J. & Krzywicki, A. [2004] “Per-turbing general uncorrelated networks,” Phys. Rev. E70, 026106.

Coelho, R., Neda, Z., Ramasco, J. & Santos, M. [2005]“A family-network model for wealth distribution insocieties,” Physica A 353, 515-528.

Csabai, I. [1994] “1/f noise in computer network traffic,”J. Phys. A Math. Gen. 27, L417–L421.

Dorogovtsev, S. N. & Mendes, J. F. [2003] Evolution ofNetworks: From Biology to the Internet and WWW(Oxford University Press).

Doyle, D., Alderson, D., Li, L., Low, S., Roughan, M.,Shalunov, S., Tanaka, R. & Willinger, W. [2005] “The‘robust yet fragile’ nature of the Internet,” PNAS 102,14497–14502.

Duch, J. & Arenas, A. [2006] “On the universality of thescaling of fluctuations in traffic on complex networks,”physics/0602077.

Echenique, P., Gomez-Gardenes, J. & Moreno, Y. [2005]“Dynamics of jamming transitions in complex net-works,” Europhys. Lett. 71, 325.

Eisler, Z. & Kertesz, J. [2005] “Random walks on com-plex networks with inhomogeneous impact,” Phys.Rev. E 71, 057104.

Eisler, Z. & Kertesz, J. [2006] “Scaling theory of tempo-ral correlations and size dependent fluctuations in thetraded value of stocks,” Phys. Rev. E 73, 046109.

Gallos, L. K. [2004] “Random walk and trapping pro-cesses on scale-free networks,” Phys. Rev. E 70,046116.

Guimera, R., Arenas, A., Dıaz-Guilera, A. & Giralt, F.[2002a] “Dynamical properties of model communica-tion networks,” Phys. Rev. E 66, 026704.

Guimera, R., Dıaz-Guilera, A., Vega-Redondo, F.,Cabrales, A. & Arenas, A. [2002b] “Optimal networktopologies for local search with congestion,” Phys.Rev. Lett. 89, 248701.

Holme, P. [2003] “Congestion and centrality in trafficflow on complex networks,” Adv. Compl. Syst. 6, 163–176.

Jarrett, T. C., Ashton, D. J., Fricker, M. & Johnson,F. N. [2005] “Abrupt structural transitions involv-ing functionally optimal networks,” ArXiv Physicse-prints.

Kujawski, B., Rodgers, G. & Tadic, B. [2006] “Localinformation-based algorithms for packet transport incomplex networks,” in ICCS 2006, Lecture Notesin Computer Science, eds. Alexnandrov, V. et al.,Vol. 3993 (Springer, Berlin), pp. 1024–1031.

Latora, V. & Marchiori, M. [2001] “Efficient behavior ofsmall-world networks,” Phys. Rev. Lett. 87, 198701.

Latora, V. & Marchiori, M. [2004] “A measure ofcentrality based on the network efficiency,” cond-mat/0402050.

Moreno, Y., Pastor-Satorras, R., Vazquez, A. &Vespignani, A. [2003] “Critical load and congestion

instabilities in scale-free networks,” Europhys. Lett.62, 292–298.

Moreno, Y., Nekovee, M. & Pacheco, A. F. [2004]“Dynamics of rumor spreading in complex networks,”Phys. Rev. E 69, 066130.

Newman, M. [2001] “Scientific collaboration networks.ii. shortest paths, weighted networks, and centrality,”Phys. Rev. E 64, 016132.

Newman, M. E. J. [2002] “The spread of epidemic diseaseon networks,” Phys. Rev. E 66, 016128.

Newman, M. E. J. [2003] “Mixing patterns in networks,”Phys. Rev. E 67, 026126.

Noh, J. D. & Rieger, H. [2004] “Random walks on com-plex networks,” Phys. Rev. Lett. 92, 118701.

Ohira, T. & Sawatari, R. [1998] “Phase transition in acomputer network traffic model,” Phys. Rev. E 58,193–195.

Park, K. & Lai, Y.-C. [2005] “Jamming in complex gra-dient networks,” Phys. Rev. E 71, 065105.

Penna, T. [1995] Phys. Rev. E 51, R1.Rosvall, M. & Sneppen, K. [2003] “Modeling dynam-

ics of information networks,” Phys. Rev. Lett. 91,178701.

Rosvall, M., Minnhagen, P. & Sneppen, K. [2005] “Navi-gating networks with limited information,” Phys. Rev.E 71, 066111.

Sole, R. V. & Valverde, S. [2001] “Information transferand phase transitions in a model of internet traffic,”Physica A 289, 595–605.

Suvakov, M. & Tadic, B. [2006] “Transport processeson homogeneous planar graphs with scale-free loops,”Physica A 372, 354–361.

Tadic, B. & Priezzhev, V. [2002] “Voltage distributionin growing conducting networks,” Europ. Phys. J. B30, 143–146.

Tadic, B. & Rodgers, G. [2002] “Packet transport onscale-free networks,” Adv. Compl. Syst. 5, 445–456.

Tadic, B. & Thurner, S. [2004] “Information super-diffusion on structured networks,” Physica A 332,566–584.

Tadic, B., Thurner, S. & Rodgers, G. J. [2004] “Trafficon complex networks: Towards understanding globalstatistical properties from microscopic density fluctu-ations,” Phys. Rev. E 69, 036102.

Tadic, B. & Thurner, S. [2005] “Search and topologyaspects in transport on scale-free networks,” PhysicaA 346, 183–190.

Tadic, B. & Thurner, S. [2006] “Traffic noise andmaximum-flow spanning trees on growing and staticnetworks,” in ICCS 2006, Lecture Notes in Com-puter Science, eds. Alexandrov, V. et al., Vol. 3993(Springer, Berlin), pp. 1016–1023.

Tadic, B. [2001a] “Adaptive random walks on the classof Web graphs,” Europ. Phys. J. B 23, 221–228.

Tadic, B. [2001b] “Dynamics of directed graphs: Theworld-wide Web,” Physica A 293, 273–284.


Tadic, B. [2002] “Temporal fractal structures: Origin ofpower laws in the world-wide Web,” Physica A 314,278–283.

Tadic, B. [2003] “Modeling traffic of information packetson graphs with complex topology,” in ComputationalScience — ICCS 2003, Lecture Notes in ComputerScience, eds. Sloot, P. et al., Vol. 2657 (Springer),pp. 136–143.

Takayasu, M., Takayasu, H. & Sato, T. [1996] “Criticalbehaviors and 1/f noise in information traffic,” Phys-ica A 233, 824–834.

Takayasu, M., Takayasu, H. & Fukuda, K. [2000]“Dynamic phase transition observed in the internettraffic flow,” Physica A 277, 248–255.

Thurner, S., Hanel, R. & Pichler, S. [2003] “Risk trading,network topology, and banking regulation,” Quant.Fin. 3, 306–319.

Toroczkai, Z. & Bassler, K. [2004] “Jamming is limitedin scale-free systems,” Nature 428, p. 716.

Tsallis, C. & Stariolo, D. A. [1994] in Annual Reviewof Computational Physics II, ed. Stauffer, D. (WorldScientific, Singapore), pp. 343–356.

Tsallis, C. [1988] “Possible generalization of Boltzmann–Gibbs statistics,” J. Stat. Phys. 52, p. 479.

Valverde, S. & Sole, R. V. [2002] “Self-organized criti-cal traffic in parallel computer networks,” Physica A312, 636–648.

Valverde, S. & Sole, R. V. [2004] “Internet’s citical pathhorizon,” Europ. Phys. J. B 38, 245–252.

Vazquez, A., Pastor-Satorras, R. & Vespignani, A. [2002]“Large-scale topological and dynamical properties ofinternet,” Phys. Rev. E 65, 066130.

Wang, W.-X., Wang, B.-H., Yin, C.-Y., Xie, Y.-B. &Zhou, T. [2006] “Traffic dynamics based on local rout-ing protocols on a scale-free network,” Phys. Rev. E73, 026111.

Whitt, W. [2001] Stochastic-Process Limits (Springer,NY).

Willinger, W., Taqqu, M., Sherman, R. & Wilson, D.[1997] IEEE/ACM Trans. Networking 5, p. 711.

Willinger, W., Govindan, R., Jamin, S., Rexons, V. &Shenker, S. [2002] “Scaling phenomena in the inter-net: Critically examining criticality,” PNAS 99, 2573–2580.

Zhou, T., Liu, J.-G. & Wang, B.-H. [2005] “Com-ment on scientific collaboration networks. II. Short-est paths, weighted networks, and centrallity,”arXiv:physics/0511084.

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