このページは http://www.slideshare.net/shu-t/networkgrowth-rule-dependence-of-fractal-dimension-of-percolation-cluster-on-square-lattice の内容を掲載しています。

掲載を希望されないスライド著者の方は、こちらよりご連絡下さい。

3年弱前 (2013/12/31)にアップロードinテクノロジー

Our paper entitled “Network-Growth Rule Dependence of Fractal Dimension of Percolation Cluster on...

Our paper entitled “Network-Growth Rule Dependence of Fractal Dimension of Percolation Cluster on Square Lattice" was published in Journal of the Physical Society of Japan. This work was done in collaboration with Dr. Ryo Tamura (NIMS).

http://journals.jps.jp/doi/abs/10.7566/JPSJ.82.053002

NIMSの田村亮さんとの共同研究論文 “Network-Growth Rule Dependence of Fractal Dimension of Percolation Cluster on Square Lattice" が Journal of the Physical Society of Japan に掲載されました。

http://journals.jps.jp/doi/abs/10.7566/JPSJ.82.053002

- Second-Order Phase Transition in Heisenberg Model on Triangular Lattice with Competing Interactions3年弱前 by Shu Tanaka
- Interlayer-Interaction Dependence of Latent Heat in the Heisenberg Model on a Stacked Triangular Lattice with Competing Interactions3年弱前 by Shu Tanaka
- 2016/5/11 RCO Study Night「D-Waveが切り開く、量子コンピュータを活用する未来」開催の挨拶6ヶ月前 by Shu Tanaka

- Network-Growth Rule Dependence of Fractal Dimension

of Percolation Cluster on Square Lattice

Shu Tanaka and Ryo Tamura

Journal of the Physical Society of Japan 82, 053002 (2013) - Main Results

We studied per

Person-to-person distribution

c

by

ola

the

tion tr

author only. Not

ansition beha

permitted for publication for insti

vior in a net

tutional repositories or on per wor

sonal

k g

Web sites.rowth

model. We focused on network-growth rule dependence of

J. Phys. Soc. Jpn. 82 (2013) 053002

LETTERS

S. TANAKA and R. TAMURA

percolation cluster geometry.

5

5

5

5

6

6

5

1

1

5

6

(a)

(b)

1

1

1

4

4

4

2.00

1

1

4

4

4

1

2

2

4

4

4

5 Achlioptas

5

5

5

6

6

1

2

4

2

2

1.95

5

3

3

5

6

1

2

2

2

2

2

3

3

3

4

4

4

1

1

2

2

2

2

106

3

3

4

4

4

3

2

2

4

4

4

105

1.90

5

5

5

5

4

4

n p

1

2

4

2

2random

104

1

2

2

2

2

2

5

3

3

5

4

inverse

1

1

2

2 Achlioptas

2

2

3

3

3

4

4

4

103

1.85

3

3

4

4

4

101

102

103 101

102

103 101

102

103 101

102

103 101

102

103 101

102

103

-10-6

-10-4

-10-2

-1

10-6 10-4 10-2

1

R

q

p

Rp

Rp

Rp

Rp

Rp

3

2

2

4

4

4

1

2

4

2

2

Fig.- A gener

5.

(Color online)

aliz

(a)

ed net

(Upper panels)

w

Snapshots or

at the k-gro

percolation wth rule w

point for q ¼ À1 (inverseas

Achlioptas rule), À10À2, 0 (random rule), 10À5, 10À2, 1

2

2

2

2

2

and þ1 (Achlioptas rule) from left to right. The dark and light points depict elements in the percolation cluster and in the second-largest cluster, 1 1

2

2

2

2

respectively. (Lower panels) Number of elements in the percolation cluster n

constructed.

p as a function of gyradius Rp for corresponding q. The dotted lines are obtained

by least-squares estimation using Eq. (2) and the fractal dimensions D are displayed in the bottom right corner. (b) q-dependence of fractal dimension. The

dotted lines indicate the fractal dimensions for q ¼ þ1 (Achlioptas rule), q ¼ 0 (random rule), and q ¼ À1 (inverse Achlioptas rule) from top to bottom.

- As the speed of growth increases, the roughness parameter of

per

conven

c

tional olation clust

self-similar structure. er decr

The upper

eases

panels of

In this study we focused on the case of a two-dimensional

Fig. 5(a) show snapshots of the percolation cluster and the

. square lattice. To investigate the relation between the spatial

second-largest cluster at the percolation point for q ¼ À1

dimension and more detailed characteristics of percolation

- A

(inverse s the speed of g

Achlioptas rule), À10À2, 0 rowth incr

(random rule), 10À5,eases

(e.g.,

, the fr

critical

ac

exponents) tal dimension of per

for our proposed rule is a remaining

colation

10À2, and þ1 (Achlioptas rule) from left to right. The

problem. Since our rule is a general rule for many network-

clust

corresponding er incr

gyradius

eases

dependence .

of np is shown in the

growth problems, it enables us to design the nature of

lower panels of Fig. 5(a), which are obtained by calculation

percolation. In this paper, we studied the fixed q-dependence

on lattice sizes from L ¼ 64 to 1280. The dotted lines

of the percolation phenomenon. However, for instance, in a

are obtained by least-squares estimation using Eq. (2).

social network, it is possible that q changes with time. Then

Figure 5(b) shows the fractal dimension as a function of q.

it is an interesting problem to consider the percolation

The fractal dimension for the inverse Achlioptas rule and

phenomenon with a time-dependent q under our rule. We

that for the random rule are of similar value, whereas that

strongly believe that our rule will provide a greater

for the Achlioptas rule obviously differs from them. Notice

understanding of percolation and will be applied for many

that the fractal dimension of a percolation cluster at the

network-growth phenomena in nature and information

percolation point on a two-dimensional lattice is D ¼

technology.

91=48 ’ 1:896 Á Á Á3) which is almost the same value as the

Acknowledgments

The authors would like to thank Takashi Mori and

fractal dimension for q ¼ 0. This result is consistent with the

Sergio Andraus for critical reading of the manuscript. S.T. is partially supported

result shown in Fig. 4(b). If the cluster is of porous structure,

by Grand-in-Aid for JSPS Fellows (23-7601) and R.T. is partially supported

by Global COE Program ‘‘the Physical Sciences Frontier’’, the Ministry of

the fractal dimension becomes small.

Education, Culture, Sports, Science and Technology, Japan and by National

In this paper, we investigated certain geometric aspects of

Institute for Materials Science (NIMS). Numerical calculations were performed

percolation clusters under a network-growth rule in which

on supercomputers at the Institute for Solid State Physics, University of Tokyo.

the introduced parameter q assigns the rule. Since our rule

includes both the Achlioptas rule7) and the random rule

where elements are randomly connected step by step, our

1) S. Kirkpatrick: Rev. Mod. Phys. 45 (1973) 574.

rule can be regarded as a generalized network-growth rule.

2) D. Stauffer: Phys. Rep. 54 (1979) 1.

We studied the time evolution of the number of elements in

3) D. Stauffer and A. Aharony: Introduction to Percolation Theory

the largest cluster. As q increases (i.e., the rule approaches

(Taylor & Francis, London, 1994).

the Achlioptas rule), the percolation step is delayed and the

4) R. Albert and A.-L. Baraba´si: Rev. Mod. Phys. 74 (2002) 47.

time evolution of n

5) S. N. Dorogovtsev, A. V. Goltsev, and J. F. F. Mendes: Rev. Mod.

max becomes explosive. We also studied

Phys. 80 (2008) 1275.

another geometric property of the percolation cluster

6) P. Erdo¨s and A. Re´nyi: Publ. Math. Inst. Hung. Acad. Sci. 5 (1960) 17.

through ns=np, which represents the roughness of the

7) D. Achlioptas, R. M. D’Souza, and J. Spencer: Science 323 (2009)

percolation cluster. The ratio ns=np monotonically decreases

1453.

against q. From these facts, the network-growth speed and

8) R. M. Ziff: Phys. Rev. Lett. 103 (2009) 045701.

geometric properties of the percolation cluster change by

9) R. M. Ziff: Phys. Rev. E 82 (2010) 051105.

10) Y. S. Cho, J. S. Kim, J. Park, B. Kahng, and D. Kim: Phys. Rev. Lett.

tuning q. Fractal dimensions for several q’s were also

103 (2009) 135702.

calculated. We found that as q increases, the fractal

11) R. A. da Costa, S. N. Dorogovtsev, A. V. Goltsev, and J. F. F. Mendes:

dimension of percolation cluster increases. It is expected

Phys. Rev. Lett. 105 (2010) 255701.

that the fractal dimension changes continuously as a function

12) P. Grassberger, C. Christensen, G. Bizhani, S.-W. Son, and M.

of q. However, since the accuracy of the fractal dimension

Paczuski: Phys. Rev. Lett. 106 (2011) 225701.

13) H. Chae, S.-H. Yook, and Y. Kim: Phys. Rev. E 85 (2012) 051118.

shown in Fig. 5(b) is not enough to conclude the expecta-

14) O. Riordan and L. Warnke: Science 333 (2011) 322.

tion, we should calculate more larger systems with high

15) N. A. M. Arau´jo and H. J. Herrmann: Phys. Rev. Lett. 105 (2010)

accuracy.

035701.

053002-4

#2013 The Physical Society of Japan - Background

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

low density

percolation

high density

point

Materials Science:

electric conductivity in metal-insulator alloys

magnetic phase transition in diluted ferromagnets

Dynamic Behavior:

spreading wildfire, spreading epidemics

Interdisciplinary Science:

network science, internet search engine

Percolation transition is a continuous transition. - Background

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

Assumption: All elements are isolated in the initial state.

Initial state

select a pair

connect a

connect a

percolated

randomly.

selected pair.

selected pair.

cluster is made.

time

Assumption: Clusters are never separated.

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

In this rule, a continuous percolation transition occurs. - Background

Suppose we consider a network model on square lattice.

Assumption: All elements are isolated in the initial state.

Select two pairs.

Connect a selected

Compare the sums of

pair.

num. of elements.

(smaller sum)

4+8=12, 3+10=13

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

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

✔ 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 confirmed well.

But there are some studies which insisted “explosive percolation is

actually continuous”.

✔ 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 unified way.

Scenario A

Scenario B

There should be boundary between

Continuous transition always occurs?

continuous and discontinuous.

what happened in intermediate region?

conventional

Achlioptas

conventional

Achlioptas

rule

rule

rule

rule

discontinuous

continuous

continuous

continuous

???

???

transition

transition

transition

transition - A generalized parameter

e q 12

e q 12 + e q 13

4+8=12, 3+10=13

4+8=12, 3+10=13

e q 13

e q 12 + e q 13

4+8=12, 3+10=13

q =

: Achlioptas rule

q = 0

: random rule

q =

: inverse Achlioptas rule - Procedures of network-growth rule

Step 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

e q[n( i)+n( j)]

wij = e q[n( i)+n( j)] + e q[n( k)+n( l)]

we connect the other edge with the probability wkl = 1

wij

Step 4: We repeat step 2 and step 3 until all of the elements belong to the

same cluster.

e q 12

e q 12 + e q 13

4+8=12, 3+10=13

4+8=12, 3+10=13

e q 13

e q 12 + e q 13

4+8=12, 3+10=13 - q-dependence of nmax

maximum of the number of elemen

nmax :

ts.

1 256 x 256 square lattice

0.8

q=-∞

q=+∞

0.6

/N

n max 0.4

0.2

0

0.75

0.8

0.85

0.9

0.95

1

t

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) - Geometric quantity ns/np

np :

the number of elements in the percolated cluster.

percolated

cluster

np = 25

ns : the number of elements in contact with other clusters in

the percolated cluster.

percolated

cluster

ns = 20 - Percolation step and geometric quantity

tp the

(L) :

first step for which a percolation cluster appears.

256 x 256 square lattice

1.00

Achlioptas

0.95

random

0.90

(L) 0.85

t p 0.80

inverse Achlioptas

0.75 (b)

inverse Achlioptas

0.40

random

p/n 0.30

n s

negative q

positive q

0.20 (c)

Achlioptas

0.10

-10-6

-10-4

-10-2

-1

10-6

10-4

10-2

1

q

As q increases, the roughness parameter ns/np decreases!! - Size dependence of tp(L)

tp(L =

)

tp(L) = aL1/

1

Achlioptas

0.95

0.9

random

(L)

t pt

0.85

0.8

inverse

Achlioptas

(a)

0.75

0

400

800

1200

L

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)

Strong size dependence can be observed at intermediate positive q. - Fractal dimension

Fractal dimension: Relation between area and characteristic length

ex.) square

ex.) sphere

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

x2

x2

J. Phys. Soc. Jpn. 82 (2013) 053002

LETTERS

S. TANAKA and R. TAMURA

D = d(= 2)

D = d(= 2)

3

(a)

(b) 2.00

Achlioptas

1.95

106

105

1.90

n p

random

104

As q increases, the fractal dimension of

inverse

Achlioptas

percolation cluster increases!!

103

1.85

101

102

103 101

102

103 101

102

103 101

102

103 101

102

103 101

102

103

-10-6

-10-4

-10-2

-1

10-6 10-4 10-2

1

R

q

p

Rp

Rp

Rp

Rp

Rp

Fig. 5.

(Color online) (a) (Upper panels) Snapshots at the percolation point for q ¼ À1 (inverse Achlioptas rule), À10À2, 0 (random rule), 10À5, 10À2,

and þ1 (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 gyradius Rp for corresponding q. The dotted lines are obtained

by least-squares estimation using Eq. (2) and the fractal dimensions D are displayed in the bottom right corner. (b) q-dependence of fractal dimension. The

dotted lines indicate the fractal dimensions for q ¼ þ1 (Achlioptas rule), q ¼ 0 (random rule), and q ¼ À1 (inverse Achlioptas rule) from top to bottom.

conventional self-similar structure. The upper panels of

In this study we focused on the case of a two-dimensional

Fig. 5(a) show snapshots of the percolation cluster and the

square lattice. To investigate the relation between the spatial

second-largest cluster at the percolation point for q ¼ À1

dimension and more detailed characteristics of percolation

(inverse Achlioptas rule), À10À2, 0 (random rule), 10À5,

(e.g., critical exponents) for our proposed rule is a remaining

10À2, and þ1 (Achlioptas rule) from left to right. The

problem. Since our rule is a general rule for many network-

corresponding gyradius dependence of np is shown in the

growth problems, it enables us to design the nature of

lower panels of Fig. 5(a), which are obtained by calculation

percolation. In this paper, we studied the fixed q-dependence

on lattice sizes from L ¼ 64 to 1280. The dotted lines

of the percolation phenomenon. However, for instance, in a

are obtained by least-squares estimation using Eq. (2).

social network, it is possible that q changes with time. Then

Figure 5(b) shows the fractal dimension as a function of q.

it is an interesting problem to consider the percolation

The fractal dimension for the inverse Achlioptas rule and

phenomenon with a time-dependent q under our rule. We

that for the random rule are of similar value, whereas that

strongly believe that our rule will provide a greater

for the Achlioptas rule obviously differs from them. Notice

understanding of percolation and will be applied for many

that the fractal dimension of a percolation cluster at the

network-growth phenomena in nature and information

percolation point on a two-dimensional lattice is D ¼

technology.

91=48 ’ 1:896 Á Á Á3) which is almost the same value as the

Acknowledgments

The authors would like to thank Takashi Mori and

fractal dimension for q ¼ 0. This result is consistent with the

Sergio Andraus for critical reading of the manuscript. S.T. is partially supported

result shown in Fig. 4(b). If the cluster is of porous structure,

by Grand-in-Aid for JSPS Fellows (23-7601) and R.T. is partially supported

by Global COE Program ‘‘the Physical Sciences Frontier’’, the Ministry of

the fractal dimension becomes small.

Education, Culture, Sports, Science and Technology, Japan and by National

In this paper, we investigated certain geometric aspects of

Institute for Materials Science (NIMS). Numerical calculations were performed

percolation clusters under a network-growth rule in which

on supercomputers at the Institute for Solid State Physics, University of Tokyo.

the introduced parameter q assigns the rule. Since our rule

includes both the Achlioptas rule7) and the random rule

where elements are randomly connected step by step, our

1) S. Kirkpatrick: Rev. Mod. Phys. 45 (1973) 574.

rule can be regarded as a generalized network-growth rule.

2) D. Stauffer: Phys. Rep. 54 (1979) 1.

We studied the time evolution of the number of elements in

3) D. Stauffer and A. Aharony: Introduction to Percolation Theory

the largest cluster. As q increases (i.e., the rule approaches

(Taylor & Francis, London, 1994).

the Achlioptas rule), the percolation step is delayed and the

4) R. Albert and A.-L. Baraba´si: Rev. Mod. Phys. 74 (2002) 47.

time evolution of n

5) S. N. Dorogovtsev, A. V. Goltsev, and J. F. F. Mendes: Rev. Mod.

max becomes explosive. We also studied

Phys. 80 (2008) 1275.

another geometric property of the percolation cluster

6) P. Erdo¨s and A. Re´nyi: Publ. Math. Inst. Hung. Acad. Sci. 5 (1960) 17.

through ns=np, which represents the roughness of the

7) D. Achlioptas, R. M. D’Souza, and J. Spencer: Science 323 (2009)

percolation cluster. The ratio ns=np monotonically decreases

1453.

against q. From these facts, the network-growth speed and

8) R. M. Ziff: Phys. Rev. Lett. 103 (2009) 045701.

geometric properties of the percolation cluster change by

9) R. M. Ziff: Phys. Rev. E 82 (2010) 051105.

10) Y. S. Cho, J. S. Kim, J. Park, B. Kahng, and D. Kim: Phys. Rev. Lett.

tuning q. Fractal dimensions for several q’s were also

103 (2009) 135702.

calculated. We found that as q increases, the fractal

11) R. A. da Costa, S. N. Dorogovtsev, A. V. Goltsev, and J. F. F. Mendes:

dimension of percolation cluster increases. It is expected

Phys. Rev. Lett. 105 (2010) 255701.

that the fractal dimension changes continuously as a function

12) P. Grassberger, C. Christensen, G. Bizhani, S.-W. Son, and M.

of q. However, since the accuracy of the fractal dimension

Paczuski: Phys. Rev. Lett. 106 (2011) 225701.

13) H. Chae, S.-H. Yook, and Y. Kim: Phys. Rev. E 85 (2012) 051118.

shown in Fig. 5(b) is not enough to conclude the expecta-

14) O. Riordan and L. Warnke: Science 333 (2011) 322.

tion, we should calculate more larger systems with high

15) N. A. M. Arau´jo and H. J. Herrmann: Phys. Rev. Lett. 105 (2010)

accuracy.

035701.

053002-4

#2013 The Physical Society of Japan - Person-to-person distribution by the author only. Not permitted for publication for institutional repositories or on personal Web sites.

Snapshot

J. Phys. Soc. Jpn. 82 (2013) 053002

LETTERS

S. TANAKA and R. TAMURA

(a)

(b) 2.00

Achlioptas

1.95

106

105

1.90

n p

random

104

inverse

Achlioptas

103

1.85

101

102

103 101

102

103 101

102

103 101

102

103 101

102

103 101

102

103

-10-6

-10-4

-10-2

-1

10-6 10-4 10-2

1

R

q

p

Rp

Rp

Rp

Rp

Rp

Fig. 5.

(Color online) (a) (Upper panels) Snapshots at the percolation point for q ¼ À1 (inverse Achlioptas rule), À10À2, 0 (random rule), 10À5, 10À2,

and þ1 (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)

As q inc

Number

reases of elements in

, the rou the percolation cluster

ghness para np as

meta function

er n of gyradius

s/np dec Rp for correspon

reases!!

ding q. The dotted lines are obtained

by least-squares estimation using Eq. (2) and the fractal dimensions D are displayed in the bottom right corner. (b) q-dependence of fractal dimension. The

dotted lines indicate the fractal dimensions for q ¼ þ1 (Achlioptas rule), q ¼ 0 (random rule), and q ¼ À1 (inverse Achlioptas rule) from top to bottom.

As q increases, the fractal dimension of percolation cluster

increases!!

conventional self-similar structure. The upper panels of

In this study we focused on the case of a two-dimensional

Fig. 5(a) show snapshots of the percolation cluster and the

square lattice. To investigate the relation between the spatial

second-largest cluster at the percolation point for q ¼ À1

dimension and more detailed characteristics of percolation

(inverse Achlioptas rule), À10À2, 0 (random rule), 10À5,

(e.g., critical exponents) for our proposed rule is a remaining

10À2, and þ1 (Achlioptas rule) from left to right. The

problem. Since our rule is a general rule for many network-

corresponding gyradius dependence of np is shown in the

growth problems, it enables us to design the nature of

lower panels of Fig. 5(a), which are obtained by calculation

percolation. In this paper, we studied the fixed q-dependence

on lattice sizes from L ¼ 64 to 1280. The dotted lines

of the percolation phenomenon. However, for instance, in a

are obtained by least-squares estimation using Eq. (2).

social network, it is possible that q changes with time. Then

Figure 5(b) shows the fractal dimension as a function of q.

it is an interesting problem to consider the percolation

The fractal dimension for the inverse Achlioptas rule and

phenomenon with a time-dependent q under our rule. We

that for the random rule are of similar value, whereas that

strongly believe that our rule will provide a greater

for the Achlioptas rule obviously differs from them. Notice

understanding of percolation and will be applied for many

that the fractal dimension of a percolation cluster at the

network-growth phenomena in nature and information

percolation point on a two-dimensional lattice is D ¼

technology.

91=48 ’ 1:896 Á Á Á3) which is almost the same value as the

Acknowledgments

The authors would like to thank Takashi Mori and

fractal dimension for q ¼ 0. This result is consistent with the

Sergio Andraus for critical reading of the manuscript. S.T. is partially supported

result shown in Fig. 4(b). If the cluster is of porous structure,

by Grand-in-Aid for JSPS Fellows (23-7601) and R.T. is partially supported

by Global COE Program ‘‘the Physical Sciences Frontier’’, the Ministry of

the fractal dimension becomes small.

Education, Culture, Sports, Science and Technology, Japan and by National

In this paper, we investigated certain geometric aspects of

Institute for Materials Science (NIMS). Numerical calculations were performed

percolation clusters under a network-growth rule in which

on supercomputers at the Institute for Solid State Physics, University of Tokyo.

the introduced parameter q assigns the rule. Since our rule

includes both the Achlioptas rule7) and the random rule

where elements are randomly connected step by step, our

1) S. Kirkpatrick: Rev. Mod. Phys. 45 (1973) 574.

rule can be regarded as a generalized network-growth rule.

2) D. Stauffer: Phys. Rep. 54 (1979) 1.

We studied the time evolution of the number of elements in

3) D. Stauffer and A. Aharony: Introduction to Percolation Theory

the largest cluster. As q increases (i.e., the rule approaches

(Taylor & Francis, London, 1994).

the Achlioptas rule), the percolation step is delayed and the

4) R. Albert and A.-L. Baraba´si: Rev. Mod. Phys. 74 (2002) 47.

time evolution of n

5) S. N. Dorogovtsev, A. V. Goltsev, and J. F. F. Mendes: Rev. Mod.

max becomes explosive. We also studied

Phys. 80 (2008) 1275.

another geometric property of the percolation cluster

6) P. Erdo¨s and A. Re´nyi: Publ. Math. Inst. Hung. Acad. Sci. 5 (1960) 17.

through ns=np, which represents the roughness of the

7) D. Achlioptas, R. M. D’Souza, and J. Spencer: Science 323 (2009)

percolation cluster. The ratio ns=np monotonically decreases

1453.

against q. From these facts, the network-growth speed and

8) R. M. Ziff: Phys. Rev. Lett. 103 (2009) 045701.

geometric properties of the percolation cluster change by

9) R. M. Ziff: Phys. Rev. E 82 (2010) 051105.

10) Y. S. Cho, J. S. Kim, J. Park, B. Kahng, and D. Kim: Phys. Rev. Lett.

tuning q. Fractal dimensions for several q’s were also

103 (2009) 135702.

calculated. We found that as q increases, the fractal

11) R. A. da Costa, S. N. Dorogovtsev, A. V. Goltsev, and J. F. F. Mendes:

dimension of percolation cluster increases. It is expected

Phys. Rev. Lett. 105 (2010) 255701.

that the fractal dimension changes continuously as a function

12) P. Grassberger, C. Christensen, G. Bizhani, S.-W. Son, and M.

of q. However, since the accuracy of the fractal dimension

Paczuski: Phys. Rev. Lett. 106 (2011) 225701.

13) H. Chae, S.-H. Yook, and Y. Kim: Phys. Rev. E 85 (2012) 051118.

shown in Fig. 5(b) is not enough to conclude the expecta-

14) O. Riordan and L. Warnke: Science 333 (2011) 322.

tion, we should calculate more larger systems with high

15) N. A. M. Arau´jo and H. J. Herrmann: Phys. Rev. Lett. 105 (2010)

accuracy.

035701.

053002-4

#2013 The Physical Society of Japan - Main Results

We studied per

Person-to-person distribution

c

by

ola

the

tion tr

author only. Not

ansition beha

permitted for publication for insti

vior in a net

tutional repositories or on per wor

sonal

k g

Web sites.rowth

model. We focused on network-growth rule dependence of

J. Phys. Soc. Jpn. 82 (2013) 053002

LETTERS

S. TANAKA and R. TAMURA

percolation cluster.

5

5

5

5

6

6

5

1

1

5

6

(a)

(b)

1

1

1

4

4

4

2.00

1

1

4

4

4

1

2

2

4

4

4

5 Achlioptas

5

5

5

6

6

1

2

4

2

2

1.95

5

3

3

5

6

1

2

2

2

2

2

3

3

3

4

4

4

1

1

2

2

2

2

106

3

3

4

4

4

3

2

2

4

4

4

105

1.90

5

5

5

5

4

4

n p

1

2

4

2

2random

104

1

2

2

2

2

2

5

3

3

5

4

inverse

1

1

2

2 Achlioptas

2

2

3

3

3

4

4

4

103

1.85

3

3

4

4

4

101

102

103 101

102

103 101

102

103 101

102

103 101

102

103 101

102

103

-10-6

-10-4

-10-2

-1

10-6 10-4 10-2

1

R

q

p

Rp

Rp

Rp

Rp

Rp

3

2

2

4

4

4

1

2

4

2

2

Fig.- A gener

5.

(Color online)

aliz

(a)

ed net

(Upper panels)

w

Snapshots or

at the k-gro

percolation wth rule w

point for q ¼ À1 (inverseas

Achlioptas rule), À10À2, 0 (random rule), 10À5, 10À2, 1

2

2

2

2

2

and þ1 (Achlioptas rule) from left to right. The dark and light points depict elements in the percolation cluster and in the second-largest cluster, 1 1

2

2

2

2

respectively. (Lower panels) Number of elements in the percolation cluster n

constructed.

p as a function of gyradius Rp for corresponding q. The dotted lines are obtained

by least-squares estimation using Eq. (2) and the fractal dimensions D are displayed in the bottom right corner. (b) q-dependence of fractal dimension. The

dotted lines indicate the fractal dimensions for q ¼ þ1 (Achlioptas rule), q ¼ 0 (random rule), and q ¼ À1 (inverse Achlioptas rule) from top to bottom.

- As the speed of growth increases, the roughness parameter of

per

conven

c

tional olation clust

self-similar structure. er decr

The upper

eases

panels of

In this study we focused on the case of a two-dimensional

Fig. 5(a) show snapshots of the percolation cluster and the

. square lattice. To investigate the relation between the spatial

second-largest cluster at the percolation point for q ¼ À1

dimension and more detailed characteristics of percolation

- A

(inverse s the speed of g

Achlioptas rule), À10À2, 0 rowth incr

(random rule), 10À5,eases

(e.g.,

, the fr

critical

ac

exponents) tal dimension of per

for our proposed rule is a remaining

colation

10À2, and þ1 (Achlioptas rule) from left to right. The

problem. Since our rule is a general rule for many network-

clust

corresponding er incr

gyradius

eases

dependence .

of np is shown in the

growth problems, it enables us to design the nature of

lower panels of Fig. 5(a), which are obtained by calculation

percolation. In this paper, we studied the fixed q-dependence

on lattice sizes from L ¼ 64 to 1280. The dotted lines

of the percolation phenomenon. However, for instance, in a

are obtained by least-squares estimation using Eq. (2).

social network, it is possible that q changes with time. Then

Figure 5(b) shows the fractal dimension as a function of q.

it is an interesting problem to consider the percolation

The fractal dimension for the inverse Achlioptas rule and

phenomenon with a time-dependent q under our rule. We

that for the random rule are of similar value, whereas that

strongly believe that our rule will provide a greater

for the Achlioptas rule obviously differs from them. Notice

understanding of percolation and will be applied for many

that the fractal dimension of a percolation cluster at the

network-growth phenomena in nature and information

percolation point on a two-dimensional lattice is D ¼

technology.

91=48 ’ 1:896 Á Á Á3) which is almost the same value as the

Acknowledgments

The authors would like to thank Takashi Mori and

fractal dimension for q ¼ 0. This result is consistent with the

Sergio Andraus for critical reading of the manuscript. S.T. is partially supported

result shown in Fig. 4(b). If the cluster is of porous structure,

by Grand-in-Aid for JSPS Fellows (23-7601) and R.T. is partially supported

by Global COE Program ‘‘the Physical Sciences Frontier’’, the Ministry of

the fractal dimension becomes small.

Education, Culture, Sports, Science and Technology, Japan and by National

In this paper, we investigated certain geometric aspects of

Institute for Materials Science (NIMS). Numerical calculations were performed

percolation clusters under a network-growth rule in which

on supercomputers at the Institute for Solid State Physics, University of Tokyo.

the introduced parameter q assigns the rule. Since our rule

includes both the Achlioptas rule7) and the random rule

where elements are randomly connected step by step, our

1) S. Kirkpatrick: Rev. Mod. Phys. 45 (1973) 574.

rule can be regarded as a generalized network-growth rule.

2) D. Stauffer: Phys. Rep. 54 (1979) 1.

We studied the time evolution of the number of elements in

3) D. Stauffer and A. Aharony: Introduction to Percolation Theory

the largest cluster. As q increases (i.e., the rule approaches

(Taylor & Francis, London, 1994).

the Achlioptas rule), the percolation step is delayed and the

4) R. Albert and A.-L. Baraba´si: Rev. Mod. Phys. 74 (2002) 47.

time evolution of n

5) S. N. Dorogovtsev, A. V. Goltsev, and J. F. F. Mendes: Rev. Mod.

max becomes explosive. We also studied

Phys. 80 (2008) 1275.

another geometric property of the percolation cluster

6) P. Erdo¨s and A. Re´nyi: Publ. Math. Inst. Hung. Acad. Sci. 5 (1960) 17.

through ns=np, which represents the roughness of the

7) D. Achlioptas, R. M. D’Souza, and J. Spencer: Science 323 (2009)

percolation cluster. The ratio ns=np monotonically decreases

1453.

against q. From these facts, the network-growth speed and

8) R. M. Ziff: Phys. Rev. Lett. 103 (2009) 045701.

geometric properties of the percolation cluster change by

9) R. M. Ziff: Phys. Rev. E 82 (2010) 051105.

10) Y. S. Cho, J. S. Kim, J. Park, B. Kahng, and D. Kim: Phys. Rev. Lett.

tuning q. Fractal dimensions for several q’s were also

103 (2009) 135702.

calculated. We found that as q increases, the fractal

11) R. A. da Costa, S. N. Dorogovtsev, A. V. Goltsev, and J. F. F. Mendes:

dimension of percolation cluster increases. It is expected

Phys. Rev. Lett. 105 (2010) 255701.

that the fractal dimension changes continuously as a function

12) P. Grassberger, C. Christensen, G. Bizhani, S.-W. Son, and M.

of q. However, since the accuracy of the fractal dimension

Paczuski: Phys. Rev. Lett. 106 (2011) 225701.

13) H. Chae, S.-H. Yook, and Y. Kim: Phys. Rev. E 85 (2012) 051118.

shown in Fig. 5(b) is not enough to conclude the expecta-

14) O. Riordan and L. Warnke: Science 333 (2011) 322.

tion, we should calculate more larger systems with high

15) N. A. M. Arau´jo and H. J. Herrmann: Phys. Rev. Lett. 105 (2010)

accuracy.

035701.

053002-4

#2013 The Physical Society of Japan - Thank you !

Shu Tanaka and Ryo Tamura

Journal of the Physical Society of Japan 82, 053002 (2013)