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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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
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)
in real systems?
Achlioptas: k-sat problems
Demanded by energy-saving in an
ad hoc network, every node has a limited
transmission range, could not
connect to all others directly.
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
Effect of transmitting range
increases transmitting radius
increases (Nodes can not be recharged)
(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
Demanded by energy-saving in an
ad hoc network, every node has a limited
transmission range, could not
connect all others directly.
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
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.
Radhika Ranjan Roy, Gravity Mobility
Handbook of Mobile Ad Hoc Networks for Mobility Models
Part 2, 443-482 (2011)
1. Gravitation rule
2. Topological connection within transmission range
3. Gravitation rule within transmission range
Continuous percolation transition/ “explosive percolation”？
with the rules depending on geometric distance
With maximum gravitation：
With minimum gravitation：
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.
[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!
(0.36, 1.5) taking geometric distance into account.