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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

Outlines depending on geometric distance on a

- 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 depending on geometric distance on a ：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 depending on geometric distance on a ：Product Rule

- The background of explosive percolation
in real systems?

Achlioptas: k-sat problems

Background depending on geometric distance on a ：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 depending on geometric distance on a

traditional communication network mobile ad hoc network

Introduction to MANET depending on geometric distance on a

- 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 depending on geometric distance on a

increases transmitting radius

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

- Network output: decreases
(MAC mechanism)

Decrease transmitting radius depending on geometric distance on a

network breaks into

pieces

Effect of transmitting radiusContradiction and equilibrium depending on geometric distance on a

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 depending on geometric distance on a caling behavior of critical connectivity

r

0.21

0.13

0.09

0.065

0.037

S depending on geometric distance on a caling behavior of critical connectivity

Background depending on geometric distance on a ：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 depending on geometric distance on a ：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 depending on geometric distance on a ：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.

- Karbovski depending on geometric distance on a

Gravity model in MANET depending on geometric distance on a

- Gravity model in MANET
Radhika Ranjan Roy, Gravity Mobility

Handbook of Mobile Ad Hoc Networks for Mobility Models

Part 2, 443-482 (2011)

motivation depending on geometric distance on a

- 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 depending on geometric distance on a

- 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 depending on geometric distance on a
with the rules depending on geometric distance

With depending on geometric distance on a 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 depending on geometric distance on a

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 depending on geometric distance on a

=0.006, =0.17N=L*L， L=128， T0=0.826

Scaling relation of percolation probability C(T,d)

With mim Grav. depending on geometric distance on a ：

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 depending on geometric distance on a

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. depending on geometric distance on a ：

Min. Grav.：

Model 3：gravitation rules inside transmission rangesInside a transmission range r

Purple circle: transmission circle

Gravitation rule inside a transmission range depending on geometric distance on a ：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 depending on geometric distance on a ：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

Finite size scaling transformation depending on geometric distance on a ：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 depending on geometric distance on a

g/n=1-b/n.

Conclusions depending on geometric distance on a

- 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.

Conclusions depending on geometric distance on a

- 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.

参考文献 depending on geometric distance on a

[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 depending on geometric distance on a !

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

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