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, China2.Research Center of complex system sciences of Shanghai University of Science and Technology, Shanghai, China
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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
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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.
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Background : Product Rule
The background of explosive percolation
in real systems?
Achlioptas: k-sat problems
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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.
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Introduction to MANET
traditional communication network mobile ad hoc network
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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
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increases transmitting radius
Interference between nodes: increases
Energy consumption: increases (Nodes can not be
recharged) Network output:
decreases(MAC mechanism)
Effect of transmitting range
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Effect of transmitting radius
Decrease transmitting radius
network breaks into
pieces
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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.
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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
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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
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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.
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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
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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
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Karbovski
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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)
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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” ?
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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
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Link-adding percolation of networks
with the rules depending on geometric distance
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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 :
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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.
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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
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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.
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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.
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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
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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
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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
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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
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Scaling law for continuous phase transition
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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.
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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
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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.
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Thank you!
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We can shift thresholds of percolation in
(0.36, 1.5) taking geometric distance into account.
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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
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