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

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)