Link-adding percolations of networks with the rules depending on geometric distance on a two-dimensional plane

Chen-Ping Zhu1,2, Long Tao Jia1

1.Nanjing University of Aeronautics and Astronautics, Nanjing, China2.Research Center of complex system sciences of Shanghai University of Science and Technology, Shanghai, China

2

Outlines

BackgroundMotivationLink-adding percolation of networks with the ru

les depending on Generalized gravitation Topological links inside a transmission range Generalized gravitation inside a transmission rang

e

Conclusions

3

Background ： Product Rule

B ： Achlioptas link-adding process ， The product Rule. Randomly choose two candidate links, and count the masses of components M1,M2,M3,M4 ,respectively, the nodes belongs to. Link the e1 if

)4(M*)4(M)2(M*)7(M 4321

A: the rule yielding ER graph ， link two disconnected nodes arbitrarily 。

Science, Achlioptas, 323, 1453-1455(2009)

C ： phases in A, B processes. The ratio of size(mass) of giant component increase with the number of added links.

4

Background ： Product Rule

The background of explosive percolation

in real systems?

Achlioptas: k-sat problems

5

Background ： Transmitting rangeand decaying probability on geometric distance

Transmitting range of mobile ad hoc networks(MANET)

Demanded by energy-saving in an ad hoc network, every node has a limited transmission range, could not connect to all others directly.

Linking probability decays with geometric distance--gravitation modelsTo link or not depending real distance in most of practical networks. Generally speaking, connecting probability decays as the distance.

G.Li, H.E.Stanley , PRL 104(018701). 2010. Yanqing.Hu, Zengru.Di , arxiv. 2010.

6

Introduction to MANET

traditional communication network mobile ad hoc network 　　　

　

7

Introduction to MANET

A mobile ad hoc network is a collection of nodes. Wireless communication among nodes works over possibly multi-hop paths without the help of any infrastructure such as base stations.

Ad hoc network: infrastructureless, peer-to-peer network, multi-hop, self-organized dynamically, energy-limited

8

increases transmitting radius

Interference between nodes: increases

Energy consumption: increases (Nodes can not be

recharged) Network output:

decreases(MAC mechanism)

Effect of transmitting range

9

Effect of transmitting radius

Decrease transmitting radius

network breaks into

pieces

10

Contradiction and equilibrium

A contradiction between global connectivity

and energy-saving (life-time)!

An equilibrium between both sides is demanded,

which asks transmission radius r and occupation

density of nodes adapt to each other.

11

Scaling behavior of critical connectivity

r

0.21

0.13

0.09

0.065

0.037

c

02r

03r

04r

06r

05r

0.01 0.1

0.0

0.2

0.4

0.6

0.8

1.0

r=2r0

r=3r0

r=4r0

r=5r0

r=6r0

analysis

12

Scaling behavior of critical connectivity

~ ( )f R

0 0( )R r r r

0.49

-6.0 -5.5 -5.0 -4.5 -4.0 -3.5 -3.0 -2.5 -2.0 -1.5 -1.0 -0.5-0.1

0.00.0

0.1

0.20.2

0.3

0.40.4

0.5

0.60.6

0.7

0.80.8

0.9

1.01.0

1.1

0.1 0.2 0.3 0.4-0.2

0.00.0

0.2

0.40.4

0.6

0.80.8

1.0

1.21.2

1.4

ln

r=2r0

r=3r0

r=4r0

r=5r0

r=6r0

R=1 R=2 R=3 R=4 R=5

r0=1

N=40000

N=3600 N=10000 N=40000

13

Background ： Transmitting rangeand geometric distance

Transmitting range of mobile ad hoc networks(MANET)

Demanded by energy-saving in an ad hoc network, every node has a limited transmission range, could not connect all others directly.

Linking probability decays with geometric distance--gravitation modelsTo link or not depending real distance in most of practical networks. Generally speaking, connecting probability decays as the distance.

G.Li, H.E.Stanley , PRL 104(018701). 2010. Yanqing.Hu, Zengru.Di , arxiv. 2010.

14

Background ： linking probability decays as the distance with the power d

G.Li, H.E.Stanley , PRL 104(018701). 2010. Cost model

d in the present work, adjustable

15

Background ： gravitation models

A tool for analyzing bilateral trading, traffic flux The scale of bilateral trading is proportional to gross

economic quantity of each side, inversely proportional to the distance between them.

ij

jiij R

YYKM

J. E. Anderson, The American Economic Review, 1979

Deardorff, A.V., NBER Working Paper 5377.1995.

J.H. Bergstrand ., The review of economics and statistics.1985.

E Helpman, PR Krugman , MIT press Cambridge.1985.

J.Tinbergen, 1962. P, Pöyhönen, Weltwirtschaftliches Archiv, 1963

16

Karbovski

17

Gravity model in MANET

Gravity model in MANETRadhika Ranjan Roy, Gravity Mobility

Handbook of Mobile Ad Hoc Networks for Mobility Models

Part 2, 443-482 (2011)

18

motivation

What effect will be caused when Product Rule is combined with the ingredient of distance? 1. Gravitation rule

2. Topological connection within transmission range

3. Gravitation rule within transmission range

Continuous percolation transition/ “explosive percolation” ？

19

Denote quantities

N: number of nodes ； N=L*L; L length of the lattice;

T: number of total links /N ； R : geometric distance between nodes;M: mass of a component;d: adjustable parameter ； r: transmission radius ；C: the ratio of the largest component, M/N; Tc: transition point

20

Link-adding percolation of networks

with the rules depending on geometric distance

21

Model 1 ： decaying on distance to the power of d (generalized gravitation)

Produce 2 links just as the PR, calculate the masses of components that 4 nodes belong to

Question:Facilitate/prohibit percolation?

dd RR 34

43

12

21 M*MM*MWith maximum gravitation ：

dd RR 34

43

12

21 M*MM*MWith minimum gravitation ：

22

The rule with minimum gravitation

With minimum gravitation ， percolation Probability decays as the d power of distance. Inset: Tc vs. d N=128*128. d: 0-50. 100 realizations

percolation goes towards of ER when d ---> inf.

23

With maximum gravitation

Scaling relation of percolation probability C(T,d)

-α 0

0

T TC~d [ ]

TF d

=0.006, =0.17 N=L*L ， L=128 ， T0=0.826

24

Model 2 ： topological linking within transmission range (radius r)

Purple circle ： transmission range

3 41 2

12 34

M *MM *Md dR R

Let d=0 for

Inside a given transmission range

)5(M*)3(M)3(M*)1(M 4321 With mim Grav. ：

With max Grav. ：

)5(M*)3(M)3(M*)1(M 4321

gravitation rule.

25

Topological linking inside a transmission range

Mam. Grav.Mim. Grav.

With the constraint of limited transmission ranges ， no scaling relation is found out for linking two node topologically without decaying with distance. It constraint from r becomes weaker (r increases), mim. Grav. Goes towards PR.

26

Model 3 ： gravitation rules inside transmission ranges

Inside a transmission range r

dd RR 34

43

12

21 M*MM*M

dd RR 34

43

12

21 M*MM*M

Max. Grav. ：

Min. Grav. ：Purple circle: transmission circle

27

Gravitation rule inside a transmission range ： max. grav.

Given r ， for diff. r ，select links with the rule of min. grav.,scaling relation exists, for r=(3,8)

]r

rr*T[r~C

0

0

H

=0.1 ，， d=2 ， N=L*L ， L=128 ， r0=2

28

Gravitation rule inside a transmission range ： min. grav.

Given r ， for diff. d ，select links with the rule of min.grav.,

scaling relation exists

0

0

T TC ~ [ ]

2 T

dG d

，， r=5r0 ， L=128 ，N=L*L ， T0=3

29

Finite size scaling transformation ：scaling law for continuous phase transition (min.grav.) With a given transmission radius r and distance-decaying power d

F.Radicchi, PRL, 103,168701,(2009)

]N*)T-T[(N~C /1C

/ Q ]N*)T-T[(N~χ /1C

/ Z

22 CCN

Scaling law for continuous phase transition

30

Scaling law for continuous phase transition

31

Conclusions

Based on real backgrounds ： gravitation rules, cost models, MANET ， we extend the Product Rule. We realized the crossover from continuous percolation of ER graphs to the explosive percolation with minimum gravitation rule.

Extend PR ， set up 3 types of models----gravitation rules, topological linking inside limited transmission ranges, and the combination of both, test the effects with selective preferences of maximum gravitation and minimum gravitation, respectively.

32

Conclusions

]r

rr*T[r~C

0

0

H

]T

TT[

2~C

0

0

Gd

-α 0

0

T TC~d [ d ]

TF

]N*)T-T[(N~C /1C

/ Q

]N*)T-T[(N~χ /1C

/ Z A scaling law for link-adding process with min. grav. rule is found with varying r and d, which suggests a continuous phase transition.

We can shift thresholds of percolation in (0.36, 1.5) taking geometric distance into account.

5 scaling relations are found with numerical simulations

33

参考文献[1] D. Achlioptas. R. M. D’Souza. and J. Spencer, “Explosive Percolation in Random

Networks”, Science, vol. 323, pp. 1453-1455, Mar. 2009.[2] R. M. Ziff, “Explosive Growth in Biased Dynamic Percolation on Two-Dimensional

Regular Lattice Networks”, Phys. Rev. Lett, vol. 103, pp. 045701(1)-(4), Jul. 2009.[3] Y. S. Cho. et al, “Percolation Transitions in Scale-Free Networks under the

Achlioptas Process”, Phys. Rev. Lett, vol. 103, pp. 135702(1)-(4), Sep. 2009.[4] F. Radicchi and S. Fortunato, “Explosive Percolation in Scale-Free Networks”,

Phys. Rev Lett, vol. 103, pp. 168701(1)-168701(4), Oct. 2009.[5] Friedman EJ, Landsberg AS, “Construction and Analysis of Random Networks with

Explosive Percolation”, Phys. Rev Lett, vol. 103, 255701, Dec. 2009.[6] D'Souza RM, Mitzenmacher M, “Local Cluster Aggregation Models of Explosive

Percolation”, Phys. Rev Lett, vol. 104, 195702, May. 2010.[7] Moreira AA, Oliveira EA, et al. “Hamiltonian approach for explosive percolation”,

Physical Review E, vol. 81, 040101, Apr. 2010.[8] Araujo NAM, Herrmann HJ, “Explosive Percolation via Control of the Largest

Cluster”, Phys. Rev. Lett, vol. 105, 035701, Jul. 2010.

34

Thank you!

35

We can shift thresholds of percolation in

(0.36, 1.5) taking geometric distance into account.

36

0 10000 20000 30000 40000 50000 60000 70000

1.38

1.40

1.42

1.44

1.46

1.48

Tc

N

Tc Fit of Tc

Equation y = a + b*x^c

Adj. R-Squar 0.98485

Value Standard Err

Tc a 1.54044 0.00819

Tc b -0.6337 0.04526

Tc c -0.2 0

/1)( bNTcNTc

37