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Link-adding percolations of networks with the rules depending on geometric distance on a two-dimensional plane. Chen-Ping Zhu 1,2 , Long Tao Jia 1 1.Nanjing University of Aeronautics and Astronautics, Nanjing, China 2.Research Center of complex system sciences of Shanghai University of

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slide1

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, China

2.Research Center of complex system sciences of Shanghai University of

Science and Technology, Shanghai, China

outlines
Outlines
  • Background
  • Motivation
  • Link-adding percolation of networks with the rules depending on
    • Generalized gravitation
    • Topological links inside a transmission range
    • Generalized gravitation inside a transmission range
  • Conclusions
background product rule
Background:Product Rule

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

B:Achlioptaslink-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

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

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

background product rule1
Background:Product Rule
  • The background of explosive percolation

in real systems?

Achlioptas: k-sat problems

background transmitting range and decaying probability on geometric distance
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 models

To link or not depending real distance in most of practical networks. Generally speaking, connecting probability decays as the distance.

Yanqing.Hu, Zengru.Di , arxiv.2010.

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

introduction to manet
Introduction to MANET

traditional communication network mobile ad hoc network

introduction to manet1
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
slide8

Effect of transmitting range

increases transmitting radius

  • Interference between nodes: increases
  • Energy consumption:

increases (Nodes can not be recharged)

  • Network output: decreases

(MAC mechanism)

contradiction and equilibrium
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.

s caling behavior of critical connectivity
Scaling behavior of critical connectivity

r

0.21

0.13

0.09

0.065

0.037

background transmitting range and geometric distance
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 models

To link or not depending real distance in most of practical networks. Generally speaking, connecting probability decays as the distance.

Yanqing.Hu, Zengru.Di , arxiv.2010.

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

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

a d in the present work, adjustable

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

Cost model

background gravitation models
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.

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

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

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

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

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

gravity model in manet
Gravity model in MANET
  • Gravity model in MANET

Radhika Ranjan Roy, Gravity Mobility

Handbook of Mobile Ad Hoc Networks for Mobility Models

Part 2, 443-482 (2011)

motivation
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”?

denote quantities
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
slide20
Link-adding percolation of networks

with the rules depending on geometric distance

model 1 decaying on distance to the power of d generalized gravitation

With maximum gravitation:

With minimum gravitation:

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?

the rule with minimum gravitation
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.

with maximum gravitation
With maximum gravitation

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

Scaling relation of percolation probability C(T,d)

model 2 topological linking within transmission range radius r

With mim Grav.:

With max Grav.:

Let d=0 for

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

Inside a given transmission range

gravitation rule.

Purple circle:transmission range

topological linking inside a transmission range
Topological linking inside a transmission range

Mim. Grav.

Mam. 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.

model 3 gravitation rules inside transmission ranges

Max. Grav.:

Min. Grav.:

Model 3:gravitation rules inside transmission ranges

Inside a transmission range r

Purple circle: transmission circle

gravitation rule inside a transmission range max grav
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)

d=0.1,h=0.1,d=2,

N=L*L, L=128,r0=2

gravitation rule inside a transmission range min grav
Gravitation rule inside a transmission range:min. grav.

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

scaling relation exists

f=0.23,w=-0.01,r=5r0,L=128,N=L*L,T0=3

slide29

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

Scaling law for continuous phase transition

g/n=1-b/n.

1/n=0.2, b/n=0.005, g/n=0.995

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

conclusions
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.
conclusions1
Conclusions
  • 5 scaling relations are found with numerical simulations
  • A scaling law for link-adding process with min. grav. rule is found with varying r and d, which suggests a continuous phase transition. g/n=1-b/n.
  • We can shift thresholds of percolation in (0.36, 1.5) taking geometric distance into account.
slide33
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.

slide35
We can shift thresholds of percolation in

(0.36, 1.5) taking geometric distance into account.