Network-Growth Rule Dependence of Fractal Dimension of Percolation Cluster on Square Lattice

Network-Growth Rule Dependence of Fractal Dimension of Percolation Cluster on Square Lattice

Shu Tanaka and Ryo TamuraJournal of the Physical Society of Japan 82, 053002 (2013)

Main ResultsWe studied percolation transition behavior in a network growth model. We focused on network-growth rule dependence of percolation cluster geometry.

- A generalized network-growth rule was constructed.

- As the speed of growth increases, the roughness parameter of percolation cluster decreases.

- As the speed of growth increases, the fractal dimension of percolation cluster increases.

Person-to-person distribution by the author only. Not permitted for publication for institutional repositories or on personal Web sites.

conventional self-similar structure. The upper panels ofFig. 5(a) show snapshots of the percolation cluster and thesecond-largest cluster at the percolation point for q ! "1(inverse Achlioptas rule), "10"2, 0 (random rule), 10"5,10"2, and #1 (Achlioptas rule) from left to right. Thecorresponding gyradius dependence of np is shown in thelower panels of Fig. 5(a), which are obtained by calculationon lattice sizes from L ! 64 to 1280. The dotted linesare obtained by least-squares estimation using Eq. (2).Figure 5(b) shows the fractal dimension as a function of q.The fractal dimension for the inverse Achlioptas rule andthat for the random rule are of similar value, whereas thatfor the Achlioptas rule obviously di!ers from them. Noticethat the fractal dimension of a percolation cluster at thepercolation point on a two-dimensional lattice is D !91=48 ’ 1:896 $ $ $3) which is almost the same value as thefractal dimension for q ! 0. This result is consistent with theresult shown in Fig. 4(b). If the cluster is of porous structure,the fractal dimension becomes small.

In this paper, we investigated certain geometric aspects ofpercolation clusters under a network-growth rule in whichthe introduced parameter q assigns the rule. Since our ruleincludes both the Achlioptas rule7) and the random rulewhere elements are randomly connected step by step, ourrule can be regarded as a generalized network-growth rule.We studied the time evolution of the number of elements inthe largest cluster. As q increases (i.e., the rule approachesthe Achlioptas rule), the percolation step is delayed and thetime evolution of nmax becomes explosive. We also studiedanother geometric property of the percolation clusterthrough ns=np, which represents the roughness of thepercolation cluster. The ratio ns=np monotonically decreasesagainst q. From these facts, the network-growth speed andgeometric properties of the percolation cluster change bytuning q. Fractal dimensions for several q’s were alsocalculated. We found that as q increases, the fractaldimension of percolation cluster increases. It is expectedthat the fractal dimension changes continuously as a functionof q. However, since the accuracy of the fractal dimensionshown in Fig. 5(b) is not enough to conclude the expecta-tion, we should calculate more larger systems with highaccuracy.

In this study we focused on the case of a two-dimensionalsquare lattice. To investigate the relation between the spatialdimension and more detailed characteristics of percolation(e.g., critical exponents) for our proposed rule is a remainingproblem. Since our rule is a general rule for many network-growth problems, it enables us to design the nature ofpercolation. In this paper, we studied the fixed q-dependenceof the percolation phenomenon. However, for instance, in asocial network, it is possible that q changes with time. Thenit is an interesting problem to consider the percolationphenomenon with a time-dependent q under our rule. Westrongly believe that our rule will provide a greaterunderstanding of percolation and will be applied for manynetwork-growth phenomena in nature and informationtechnology.

Acknowledgments The authors would like to thank Takashi Mori andSergio Andraus for critical reading of the manuscript. S.T. is partially supportedby Grand-in-Aid for JSPS Fellows (23-7601) and R.T. is partially supportedby Global COE Program ‘‘the Physical Sciences Frontier’’, the Ministry ofEducation, Culture, Sports, Science and Technology, Japan and by NationalInstitute for Materials Science (NIMS). Numerical calculations were performedon supercomputers at the Institute for Solid State Physics, University of Tokyo.

1) S. Kirkpatrick: Rev. Mod. Phys. 45 (1973) 574.2) D. Stau!er: Phys. Rep. 54 (1979) 1.3) D. Stau!er and A. Aharony: Introduction to Percolation Theory

(Taylor & Francis, London, 1994).4) R. Albert and A.-L. Barabasi: Rev. Mod. Phys. 74 (2002) 47.5) S. N. Dorogovtsev, A. V. Goltsev, and J. F. F. Mendes: Rev. Mod.

Phys. 80 (2008) 1275.6) P. Erdos and A. Renyi: Publ. Math. Inst. Hung. Acad. Sci. 5 (1960) 17.7) D. Achlioptas, R. M. D’Souza, and J. Spencer: Science 323 (2009)

1453.8) R. M. Zi!: Phys. Rev. Lett. 103 (2009) 045701.9) R. M. Zi!: Phys. Rev. E 82 (2010) 051105.10) Y. S. Cho, J. S. Kim, J. Park, B. Kahng, and D. Kim: Phys. Rev. Lett.

103 (2009) 135702.11) R. A. da Costa, S. N. Dorogovtsev, A. V. Goltsev, and J. F. F. Mendes:

Phys. Rev. Lett. 105 (2010) 255701.12) P. Grassberger, C. Christensen, G. Bizhani, S.-W. Son, and M.

Paczuski: Phys. Rev. Lett. 106 (2011) 225701.13) H. Chae, S.-H. Yook, and Y. Kim: Phys. Rev. E 85 (2012) 051118.14) O. Riordan and L. Warnke: Science 333 (2011) 322.15) N. A. M. Araujo and H. J. Herrmann: Phys. Rev. Lett. 105 (2010)

035701.

103

104

105

106

101 102 103

n p

Rp

101 102 103

Rp

101 102 103

Rp

101 102 103

Rp

101 102 103

Rp

101 102 103

Rp

(a)

10-6 10-4 10-2 1

q

1.85

1.90

1.95

2.00

-10-6-10-4-10-2-1

(b)

Achlioptas

inverseAchlioptas

random

Fig. 5. (Color online) (a) (Upper panels) Snapshots at the percolation point for q ! "1 (inverse Achlioptas rule), "10"2, 0 (random rule), 10"5, 10"2,and #1 (Achlioptas rule) from left to right. The dark and light points depict elements in the percolation cluster and in the second-largest cluster,respectively. (Lower panels) Number of elements in the percolation cluster np as a function of gyradiusRp for corresponding q. The dotted lines are obtainedby least-squares estimation using Eq. (2) and the fractal dimensions D are displayed in the bottom right corner. (b) q-dependence of fractal dimension. Thedotted lines indicate the fractal dimensions for q ! #1 (Achlioptas rule), q ! 0 (random rule), and q ! "1 (inverse Achlioptas rule) from top to bottom.

S. TANAKA and R. TAMURAJ. Phys. Soc. Jpn. 82 (2013) 053002 LETTERS

053002-4 #2013 The Physical Society of Japan

3

3

1

1

1

5

5

3

3

1

5

3

3

2

2

5

3

3

2

2

2

2

5

5

4

4

4

2

4

4

2

2

6

4

4

4

2

2

6

6

4

4

2

2

2

1

1

1

1

1

5

5

1

1

1

5

1

1

2

2

5

1

1

2

2

2

2

5

5

4

4

4

2

4

4

2

2

6

4

4

4

2

2

6

6

4

4

2

2

2

3

3

1

1

1

5

5

3

3

1

5

3

3

2

2

5

3

3

2

2

2

2

5

5

4

4

4

2

4

4

2

2

4

4

4

4

2

2

4

4

4

4

2

2

2

Backgroundordered state: A cluster spreads from the edge to the opposite edge.

high densitylow density percolationpoint

Materials Science: electric conductivity in metal-insulator alloys magnetic phase transition in diluted ferromagnetsDynamic Behavior: spreading wild!re, spreading epidemicsInterdisciplinary Science: network science, internet search engine

Percolation transition is a continuous transition.

BackgroundSuppose we consider a network-growth model on square lattice.

Assumption: All elements are isolated in the initial state.

Initial stateselect a pairrandomly.

connect a selected pair.

connect a selected pair.

percolated cluster is made.

timeAssumption: Clusters are never separated.

We refer to this network-growth rule as random rule.

In this rule, a continuous percolation transition occurs.

Background

Select two pairs.Compare the sums of

num. of elements.

4+8=12, 3+10=13

Connect a selected pair.

(smaller sum)

Suppose we consider a network model on square lattice.Assumption: All elements are isolated in the initial state.

Assumption: Clusters are never separated.We refer to this network-growth rule as Achlioptas rule.

In this rule, a discontinuous percolation transition occurs !?D. Achlioptas, R.M. D’Souza, J. Spencer, Science 323, 1453 (2009).

Motivation✔ Conventional percolation transition is a continuous transition. But it was reported that a discontinuous percolation transition can occur depending on network-growth rule (Achlioptas rule). This transition is called “explosive percolation transition”.

✔ Nature of explosive percolation transition has been con!rmed well. But there are some studies which insisted “explosive percolation is actually continuous”.

We consider nature of percolation transition in network-growth model.

✔ Which is the explosive percolation transition discontinuous or continuous? To understand this major challenge, we introduced a parameter which enables us to consider the network-growth model in a uni!ed way.

Scenario A

conventionalrule

Achlioptasrule

discontinuoustransition

continuoustransition

There should be boundary betweencontinuous and discontinuous.

Scenario B

conventionalrule

Achlioptasrule



Continuous transition always occurs?what happened in intermediate region?

??? ???

A generalized parameter

4+8=12, 3+10=13

4+8=12, 3+10=13

e�q�12

e�q�12 + e�q�13

4+8=12, 3+10=13

e�q�13

e�q�12 + e�q�13

q =� : Achlioptas ruleq = 0 : random ruleq = �� : inverse Achlioptas rule

Procedures of network-growth ruleStep 1: The initial state is set: All elements belong to different clusters.Step 2: Choose two different edges randomly.Step 3: We connect an edge with the probability given by

we connect the other edge with the probability Step 4: We repeat step 2 and step 3 until all of the elements belong to the same cluster.

wij =e�q[n(�i)+n(�j)]

e�q[n(�i)+n(�j)] + e�q[n(�k)+n(�l)]

wkl = 1� wij

4+8=12, 3+10=13

4+8=12, 3+10=13

4+8=12, 3+10=13

e�q�12

e�q�12 + e�q�13

e�q�13

e�q�12 + e�q�13

q-dependence of nmax

0

0.2

0.4

0.6

0.8

1

0.75 0.8 0.85 0.9 0.95 1

n max

/N

t

maximum of the number of elements.nmax :

256 x 256 square lattice

q=-∞ (inverse Achlioptas), -1, -10-1, -10 -2, -10 -3, -10 -4, 0 (random), 2.5x10 -5, 5x10 -5, 10 -4, 2x10 -4, 10 -3, 10 -2, 10 -1, and +∞ (Achlioptas)

q=-∞ q=+∞

Geometric quantity ns/np

percolated cluster

np : the number of elements in the percolated cluster.

ns : the number of elements in contact with other clusters in the percolated cluster.

percolated cluster

np = 25

ns = 20

Percolation step and geometric quantity

10-6 10-4 10-2 1q

Achlioptas

random

0.10

0.20

0.30

0.40

-10-6-10-4-10-2-1

n s/n

p

inverse Achlioptas

(b)

(c)

Achlioptasrandom

0.750.800.850.900.951.00

t p(L)

inverse Achlioptas

negative q positive q

the !rst step for which a percolation cluster appears.tp(L) :

256 x 256 square lattice

As q increases, the roughness parameter ns/np decreases!!

Size dependence of tp(L)

0.75

0.8

0.85

0.9

0.95

1

0 400 800 1200

t p(L

)

L

0.75

0.8

0.85

0.9

0.95

1

0 400 800 1200

t p(L

)

L

Achlioptas

random

inverseAchlioptas

(a)

tp(L =�)� tp(L) = aL1/�

Strong size dependence can be observed at intermediate positive q.

q=-∞ (inverse Achlioptas), -1, -10-1, -10 -2, -10 -3, -10 -4, 0 (random), 2.5x10 -5, 5x10 -5, 10 -4, 2x10 -4, 10 -3, 10 -2, 10 -1, and +∞ (Achlioptas)

Fractal dimensionFractal dimension: Relation between area and characteristic lengthex.) square

x2

D = d(= 2)

ex.) sphere

x2

D = d(= 2)3













035701.

103

104

105

106

101 102 103

n p

Rp

101 102 103

Rp

101 102 103

Rp

101 102 103

Rp

101 102 103

Rp

101 102 103

Rp

(a)

10-6 10-4 10-2 1

q

1.85

1.90

1.95

2.00

-10-6-10-4-10-2-1

(b)

Achlioptas

inverseAchlioptas

random




As q increases, the fractal dimension of percolation cluster increases!!

SnapshotPerson-to-person distribution by the author only. Not permitted for publication for institutional repositories or on personal Web sites.












035701.

103

104

105

106

101 102 103

n p

Rp

101 102 103

Rp

101 102 103

Rp

101 102 103

Rp

101 102 103

Rp

101 102 103

Rp

(a)

10-6 10-4 10-2 1

q

1.85

1.90

1.95

2.00

-10-6-10-4-10-2-1

(b)

Achlioptas

inverseAchlioptas

random




As q increases, the roughness parameter ns/np decreases!!

As q increases, the fractal dimension of percolation cluster increases!!

Main ResultsWe studied percolation transition behavior in a network growth model. We focused on network-growth rule dependence of percolation cluster.

- A generalized network-growth rule was constructed.

- As the speed of growth increases, the roughness parameter of percolation cluster decreases.

- As the speed of growth increases, the fractal dimension of percolation cluster increases.













035701.

103

104

105

106

101 102 103

n p

Rp

101 102 103

Rp

101 102 103

Rp

101 102 103

Rp

101 102 103

Rp

101 102 103

Rp

(a)

10-6 10-4 10-2 1

q

1.85

1.90

1.95

2.00

-10-6-10-4-10-2-1

(b)

Achlioptas

inverseAchlioptas

random




3

3

1

1

1

5

5

3

3

1

5

3

3

2

2

5

3

3

2

2

2

2

5

5

4

4

4

2

4

4

2

2

6

4

4

4

2

2

6

6

4

4

2

2

2

1

1

1

1

1

5

5

1

1

1

5

1

1

2

2

5

1

1

2

2

2

2

5

5

4

4

4

2

4

4

2

2

6

4

4

4

2

2

6

6

4

4

2

2

2

3

3

1

1

1

5

5

3

3

1

5

3

3

2

2

5

3

3

2

2

2

2

5

5

4

4

4

2

4

4

2

2

4

4

4

4

2

2

4

4

4

4

2

2

2

Thank you !

Shu Tanaka and Ryo TamuraJournal of the Physical Society of Japan 82, 053002 (2013)