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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 percolation transition behavior in a network growth
model. We focused on network-growth rule dependenceR.of
J. Phys. Soc. Jpn. 82 (2013) 053002
L
S. T
and
T
percolation cluster geometry.
Person-to-person distribution by the author only. Not permitted for publication for institutional repositories or on personal Web sites.

ETTERS

ANAKA

AMURA

(b)

1

5

5

1

1

5

1
2.00

5

5

(a)

5

1

1

1

1

1.95
5

3

3

3

3

1

np

104

2

2

1
1

10

2

10
Rp

3
10 10

1

2

10
Rp

1

3
10 10

2

10
Rp

3

10 101

2

10
Rp

3

10 101

2

10
Rp

1

3
10 10

2

10
Rp

10

3

2

4

2

1
1.85

3

4

1

4

4

2

2

2
-1

4

4

2
1

4

4

2

2

2

4

4

4
2

2
2

2

5

5

5

2
2

2

2

4

5

2

5

3

3

4

5

3

3

3

random

2

inverse
2 Achlioptas
2 2
10-6

10-4

10-2

1

3

3

3
À5

À2

Fig. 5. (Color online) (a) (Upper panels) Snapshots at the percolation point for q ¼ À1 (inverse Achlioptas rule), À10 , 0 (random rule), 10 , 10 ,
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.

2

1

4

2
1

2

4

2

1

4

4

2

2

2

4

4

4

4

4

1

- A generalized network-growth rule was
constructed.

4

4

4

q

À2

2

4

4

2

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

4

2

1

5
4

2

6
6

4

1

6

4

3 1.90
1

10 5

10

3

3

106

3

6
6

5 Achlioptas 5
5 5

6

4

4
2

2
2

2
2

2

2

- As the speed of growth increases, the roughness parameter of
conventional self-similar structure. The upper panels of
In this study we focused on the case of a two-dimensional
percolation the percolation cluster and the
Fig. 5(a) show snapshots of cluster decreases.square lattice. To investigate the relation between the spatial
second-largest cluster 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
corresponding gyradius dependence of np is shown in the
lower panels of Fig. 5(a), which are obtained by calculation
on lattice sizes from L ¼ 64 to 1280. The dotted lines

dimension and more detailed characteristics of percolation
(e.g., critical exponents) for our proposed rule is a remaining
problem. Since our rule is a general rule for many networkgrowth problems, it enables us to design the nature of
percolation. In this paper, we studied the fixed q-dependence
of the percolation phenomenon. However, for instance, in a

- As the speed of growth increases, the fractal dimension of percolation
cluster increases.
Background
ordered state: A cluster spreads from the edge to the opposite edge.

low density

percolation
point

high density

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
randomly.

connect a
selected pair.

connect a
selected pair.

percolated
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.
Compare the sums of
num. of elements.

Connect a selected
pair.
(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
There should be boundary between
continuous and discontinuous.

Scenario B
Continuous transition always occurs?
what happened in intermediate region?

conventional
rule
discontinuous
transition

conventional
rule
continuous
transition

???

Achlioptas
rule
continuous
transition

???

Achlioptas
rule
continuous
transition
A generalized parameter
e
e

q 12

q 12

+e

q 13

4+8=12, 3+10=13

4+8=12, 3+10=13

e
e
q=
q=0
q=

q 12

q 13

+e

: Achlioptas rule
: random rule
: inverse Achlioptas rule

q 13

4+8=12, 3+10=13
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
wij =

e
e

q[n(

q[n(

i )+n( j )]

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
e

q 12

q 12

+e

q 13

4+8=12, 3+10=13

4+8=12, 3+10=13

e
e

q 12

q 13

+e

q 13

4+8=12, 3+10=13
q-dependence of nmax
1

nmax : maximum of the number of elements.

256 x 256 square lattice

0.8

nmax/N

q=-∞

q=+∞

0.6
0.4
0.2
0
0.75

0.8

0.85

0.9

0.95

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)

1
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 (L) : the first step for which a percolation cluster appears.

tp(L)

256 x 256 square lattice
1.00
0.95
0.90
0.85
0.80
0.75

Achlioptas
random
inverse Achlioptas

(b)
inverse Achlioptas

ns/np

0.40

random

0.30

negative q

0.20
0.10

positive q

(c)
-1

Achlioptas
-2

-10

-10

-4

-10

-6

q

10

-6

10

-4

10

-2

1

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

1

)

tp (L) = aL1/
Achlioptas

tp(L)

0.95
0.9

random

0.85
0.8
0.75

inverse
Achlioptas

(a)
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

s or on personal Web sites.

0

x2
S. TANAKA and R. TAMURA

D = d(= 2)
(b)

x2
D = d(= 3
2)

2.00

Achlioptas
1.95

1.90

random
inverse
Achlioptas
3

1.85

-1

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

10-6
q

10-4

10-2

1

As q increases, the fractal dimension of
percolation cluster increases!!
Person-to-person distribution by the author only. Not permitted for publication for institutional repositories o

Snapshot
L

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

ETTERS

(a)

106

np

10 5

104

10

3
1

10

2

10
Rp

3
10 10

1

2

10
Rp

1

3
10 10

2

10
Rp

3

10 101

2

10
Rp

3

10 101

2

10
Rp

1

3
10 10

2

10
Rp

10

3

Fig. 5. (Color online) (a) (Upper panels) Snapshots at the percolation point for q ¼ À1 (inverse Achlioptas rule), À10
and þ1 (Achlioptas rule) from left to right. The dark and light points depict elements in the percolation cluster a
respectively. (Lower panels) Number of elements in the percolation cluster np as a function of gyradius Rp for correspondin
s p
by least-squares estimation using Eq. (2) and the fractal dimensions D are displayed in the bottom right corner. (b) q-depe
dotted lines indicate the fractal dimensions for q ¼ þ1 (Achlioptas rule), q ¼ 0 (random rule), and q ¼ À1 (inverse Ac

As q increases, the roughness parameter n /n decreases!!

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
Main Results
We studied percolation transition behavior in a network growth
model. We focused on network-growth rule dependenceR.of
J. Phys. Soc. Jpn. 82 (2013) 053002
L
S. T
and
T
percolation cluster.
Person-to-person distribution by the author only. Not permitted for publication for institutional repositories or on personal Web sites.

ETTERS

ANAKA

AMURA

(b)

1

5

5

1

1

5

1
2.00

5

5

(a)

5

1

1

1

1

1.95
5

3

3

3

3

1

np

104

2

2

1
1

10

2

10
Rp

3
10 10

1

2

10
Rp

1

3
10 10

2

10
Rp

3

10 101

2

10
Rp

3

10 101

2

10
Rp

1

3
10 10

2

10
Rp

10

3

2

4

2

1
1.85

3

4

1

4

4

2

2

2
-1

4

4

2
1

4

4

2

2

2

4

4

4
2

2
2

2

5

5

5

2
2

2

2

4

5

2

5

3

3

4

5

3

3

3

random

2

inverse
2 Achlioptas
2 2
10-6

10-4

10-2

1

3

3

3
À5

À2

Fig. 5. (Color online) (a) (Upper panels) Snapshots at the percolation point for q ¼ À1 (inverse Achlioptas rule), À10 , 0 (random rule), 10 , 10 ,
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.

2

1

4

2
1

2

4

2

1

4

4

2

2

2

4

4

4

4

4

1

- A generalized network-growth rule was
constructed.

4

4

4

q

À2

2

4

4

2

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

4

2

1

5
4

2

6
6

4

1

6

4

3 1.90
1

10 5

10

3

3

106

3

6
6

5 Achlioptas 5
5 5

6

4

4
2

2
2

2
2

2

2

- As the speed of growth increases, the roughness parameter of
conventional self-similar structure. The upper panels of
In this study we focused on the case of a two-dimensional
percolation the percolation cluster and the
Fig. 5(a) show snapshots of cluster decreases.square lattice. To investigate the relation between the spatial
second-largest cluster 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
corresponding gyradius dependence of np is shown in the
lower panels of Fig. 5(a), which are obtained by calculation
on lattice sizes from L ¼ 64 to 1280. The dotted lines

dimension and more detailed characteristics of percolation
(e.g., critical exponents) for our proposed rule is a remaining
problem. Since our rule is a general rule for many networkgrowth problems, it enables us to design the nature of
percolation. In this paper, we studied the fixed q-dependence
of the percolation phenomenon. However, for instance, in a

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

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

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

  • 1. 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)
  • 2. Main Results We studied percolation transition behavior in a network growth model. We focused on network-growth rule dependenceR.of J. Phys. Soc. Jpn. 82 (2013) 053002 L S. T and T percolation cluster geometry. Person-to-person distribution by the author only. Not permitted for publication for institutional repositories or on personal Web sites. ETTERS ANAKA AMURA (b) 1 5 5 1 1 5 1 2.00 5 5 (a) 5 1 1 1 1 1.95 5 3 3 3 3 1 np 104 2 2 1 1 10 2 10 Rp 3 10 10 1 2 10 Rp 1 3 10 10 2 10 Rp 3 10 101 2 10 Rp 3 10 101 2 10 Rp 1 3 10 10 2 10 Rp 10 3 2 4 2 1 1.85 3 4 1 4 4 2 2 2 -1 4 4 2 1 4 4 2 2 2 4 4 4 2 2 2 2 5 5 5 2 2 2 2 4 5 2 5 3 3 4 5 3 3 3 random 2 inverse 2 Achlioptas 2 2 10-6 10-4 10-2 1 3 3 3 À5 À2 Fig. 5. (Color online) (a) (Upper panels) Snapshots at the percolation point for q ¼ À1 (inverse Achlioptas rule), À10 , 0 (random rule), 10 , 10 , 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. 2 1 4 2 1 2 4 2 1 4 4 2 2 2 4 4 4 4 4 1 - A generalized network-growth rule was constructed. 4 4 4 q À2 2 4 4 2 -10-2 -10-4 -10-6 4 2 1 5 4 2 6 6 4 1 6 4 3 1.90 1 10 5 10 3 3 106 3 6 6 5 Achlioptas 5 5 5 6 4 4 2 2 2 2 2 2 2 - As the speed of growth increases, the roughness parameter of conventional self-similar structure. The upper panels of In this study we focused on the case of a two-dimensional percolation the percolation cluster and the Fig. 5(a) show snapshots of cluster decreases.square lattice. To investigate the relation between the spatial second-largest cluster 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 corresponding gyradius dependence of np is shown in the lower panels of Fig. 5(a), which are obtained by calculation on lattice sizes from L ¼ 64 to 1280. The dotted lines dimension and more detailed characteristics of percolation (e.g., critical exponents) for our proposed rule is a remaining problem. Since our rule is a general rule for many networkgrowth problems, it enables us to design the nature of percolation. In this paper, we studied the fixed q-dependence of the percolation phenomenon. However, for instance, in a - As the speed of growth increases, the fractal dimension of percolation cluster increases.
  • 3. Background ordered state: A cluster spreads from the edge to the opposite edge. low density percolation point high density 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.
  • 4. 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 randomly. connect a selected pair. connect a selected pair. percolated 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.
  • 5. Background Suppose we consider a network model on square lattice. Assumption: All elements are isolated in the initial state. Select two pairs. Compare the sums of num. of elements. Connect a selected pair. (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).
  • 6. 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 There should be boundary between continuous and discontinuous. Scenario B Continuous transition always occurs? what happened in intermediate region? conventional rule discontinuous transition conventional rule continuous transition ??? Achlioptas rule continuous transition ??? Achlioptas rule continuous transition
  • 7. A generalized parameter e e q 12 q 12 +e q 13 4+8=12, 3+10=13 4+8=12, 3+10=13 e e q= q=0 q= q 12 q 13 +e : Achlioptas rule : random rule : inverse Achlioptas rule q 13 4+8=12, 3+10=13
  • 8. 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 wij = e e q[n( q[n( i )+n( j )] 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 e q 12 q 12 +e q 13 4+8=12, 3+10=13 4+8=12, 3+10=13 e e q 12 q 13 +e q 13 4+8=12, 3+10=13
  • 9. q-dependence of nmax 1 nmax : maximum of the number of elements. 256 x 256 square lattice 0.8 nmax/N q=-∞ q=+∞ 0.6 0.4 0.2 0 0.75 0.8 0.85 0.9 0.95 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) 1
  • 10. 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
  • 11. Percolation step and geometric quantity tp (L) : the first step for which a percolation cluster appears. tp(L) 256 x 256 square lattice 1.00 0.95 0.90 0.85 0.80 0.75 Achlioptas random inverse Achlioptas (b) inverse Achlioptas ns/np 0.40 random 0.30 negative q 0.20 0.10 positive q (c) -1 Achlioptas -2 -10 -10 -4 -10 -6 q 10 -6 10 -4 10 -2 1 As q increases, the roughness parameter ns/np decreases!!
  • 12. Size dependence of tp(L) tp (L = 1 ) tp (L) = aL1/ Achlioptas tp(L) 0.95 0.9 random 0.85 0.8 0.75 inverse Achlioptas (a) 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.
  • 13. Fractal dimension Fractal dimension: Relation between area and characteristic length ex.) square ex.) sphere s or on personal Web sites. 0 x2 S. TANAKA and R. TAMURA D = d(= 2) (b) x2 D = d(= 3 2) 2.00 Achlioptas 1.95 1.90 random inverse Achlioptas 3 1.85 -1 -10-2 -10-4 -10-6 10-6 q 10-4 10-2 1 As q increases, the fractal dimension of percolation cluster increases!!
  • 14. Person-to-person distribution by the author only. Not permitted for publication for institutional repositories o Snapshot L J. Phys. Soc. Jpn. 82 (2013) 053002 ETTERS (a) 106 np 10 5 104 10 3 1 10 2 10 Rp 3 10 10 1 2 10 Rp 1 3 10 10 2 10 Rp 3 10 101 2 10 Rp 3 10 101 2 10 Rp 1 3 10 10 2 10 Rp 10 3 Fig. 5. (Color online) (a) (Upper panels) Snapshots at the percolation point for q ¼ À1 (inverse Achlioptas rule), À10 and þ1 (Achlioptas rule) from left to right. The dark and light points depict elements in the percolation cluster a respectively. (Lower panels) Number of elements in the percolation cluster np as a function of gyradius Rp for correspondin s p by least-squares estimation using Eq. (2) and the fractal dimensions D are displayed in the bottom right corner. (b) q-depe dotted lines indicate the fractal dimensions for q ¼ þ1 (Achlioptas rule), q ¼ 0 (random rule), and q ¼ À1 (inverse Ac As q increases, the roughness parameter n /n decreases!! 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
  • 15. Main Results We studied percolation transition behavior in a network growth model. We focused on network-growth rule dependenceR.of J. Phys. Soc. Jpn. 82 (2013) 053002 L S. T and T percolation cluster. Person-to-person distribution by the author only. Not permitted for publication for institutional repositories or on personal Web sites. ETTERS ANAKA AMURA (b) 1 5 5 1 1 5 1 2.00 5 5 (a) 5 1 1 1 1 1.95 5 3 3 3 3 1 np 104 2 2 1 1 10 2 10 Rp 3 10 10 1 2 10 Rp 1 3 10 10 2 10 Rp 3 10 101 2 10 Rp 3 10 101 2 10 Rp 1 3 10 10 2 10 Rp 10 3 2 4 2 1 1.85 3 4 1 4 4 2 2 2 -1 4 4 2 1 4 4 2 2 2 4 4 4 2 2 2 2 5 5 5 2 2 2 2 4 5 2 5 3 3 4 5 3 3 3 random 2 inverse 2 Achlioptas 2 2 10-6 10-4 10-2 1 3 3 3 À5 À2 Fig. 5. (Color online) (a) (Upper panels) Snapshots at the percolation point for q ¼ À1 (inverse Achlioptas rule), À10 , 0 (random rule), 10 , 10 , 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. 2 1 4 2 1 2 4 2 1 4 4 2 2 2 4 4 4 4 4 1 - A generalized network-growth rule was constructed. 4 4 4 q À2 2 4 4 2 -10-2 -10-4 -10-6 4 2 1 5 4 2 6 6 4 1 6 4 3 1.90 1 10 5 10 3 3 106 3 6 6 5 Achlioptas 5 5 5 6 4 4 2 2 2 2 2 2 2 - As the speed of growth increases, the roughness parameter of conventional self-similar structure. The upper panels of In this study we focused on the case of a two-dimensional percolation the percolation cluster and the Fig. 5(a) show snapshots of cluster decreases.square lattice. To investigate the relation between the spatial second-largest cluster 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 corresponding gyradius dependence of np is shown in the lower panels of Fig. 5(a), which are obtained by calculation on lattice sizes from L ¼ 64 to 1280. The dotted lines dimension and more detailed characteristics of percolation (e.g., critical exponents) for our proposed rule is a remaining problem. Since our rule is a general rule for many networkgrowth problems, it enables us to design the nature of percolation. In this paper, we studied the fixed q-dependence of the percolation phenomenon. However, for instance, in a - As the speed of growth increases, the fractal dimension of percolation cluster increases.
  • 16. Thank you ! Shu Tanaka and Ryo Tamura Journal of the Physical Society of Japan 82, 053002 (2013)