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Chen-Ping Zhu 1,2 , Long Tao Jia 1

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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

- 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

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)

- The background of explosive percolation
in real systems?

Achlioptas: k-sat problems

- 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.

traditional communication network mobile ad hoc network

- 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

Effect of transmitting range

increases transmitting radius

- Interference between nodes: increases
- Energy consumption:
increases (Nodes can not be recharged)

- Network output: decreases
(MAC mechanism)

Decrease transmitting radius

network breaks into

pieces

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.

r

0.21

0.13

0.09

0.065

0.037

- 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.

a d in the present work, adjustable

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

Cost model

- 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.

- Karbovski

- Gravity model in MANET
Radhika Ranjan Roy, Gravity Mobility

Handbook of Mobile Ad Hoc Networks for Mobility Models

Part 2, 443-482 (2011)

- 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”？

- 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

- Link-adding percolation of networks
with the rules depending on geometric distance

With maximum gravitation：

With minimum gravitation：

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

Facilitate/prohibit percolation?

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.

=0.006, =0.17N=L*L， L=128， T0=0.826

Scaling relation of percolation probability C(T,d)

With mim Grav.：

With max Grav.：

Let d=0 for

Inside a given transmission range

gravitation rule.

Purple circle：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.

Max. Grav.：

Min. Grav.：

Inside a transmission range r

Purple circle: transmission circle

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

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

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)

Scaling law for continuous phase transition

g/n=1-b/n.

- 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.

- 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.

[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.

Thank you!

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