# Critical behaviors and universality classes of percolation phase transitions
on two-dimensional square lattice

Yong Zhu  Ziqing Yang  Xin Zhang  Xiaosong Chen  State Key Laboratory of
Theoretical Physics, Institute of Theoretical Physics, Chinese Academy of
Sciences, P.O. Box 2735, Beijing 100190, China

(September 2, 2024)

###### Abstract

We have investigated both site and bond percolation on two dimensional lattice
under the random rule and the product rule respectively. With the random rule,
sites or bonds are added randomly into the lattice. From two candidates picked
randomly, the site or bond with the smaller size product of two connected
clusters is added when the product rule is taken. Not only the size of the
largest cluster but also its size jump are studied to characterize the
universality class of percolation. The finite-size scaling forms of giant
cluster size and size jump are proposed and used to determine the critical
exponents of percolation from Monte Carlo data. It is found that the critical
exponents of both size and size jump in random site percolation are equal to
that in random bond percolation. With the random rule, site and bond
percolation belong to the same universality class. We obtain the critical
exponents of the site percolation under the product rule, which are different
from that of both random percolation and the bond percolation under the
product rule. The universality class of site percolation differs different
from that of bond percolation when the product rule is used.

network, percolation phase transition

###### pacs:

64.60.ah, 64.60.De, 89.75.Da, 89.75.Hc

##  I Introduction

Traditionally, percolation phase transitions are considered to be continuous
in various networks. However, Achlioptas et al. Achlioptas  concluded that the
percolation transition in random network is discontinuous under the Achlioptas
process (AP) in which the growth of large clusters is suppressed. During the
evolution of a network under AP, edges are added into the network under
product rule that the edge with minimum product of cluster sizes is connected
from two randomly picked unoccupied edges. The percolation transitions under
product rule in two-dimensional square lattice ziff09 ; ziff10 ; Rad10  and
scale-free networks Rad10 ; ChoP09 ; Rad09  are also investigated and argued
to be discontinuous. Later, a series of investigations Costa10 ; Lee11 ;
Grassberger11 ; Riordan11 ; Liu11 ; Fan12  showed that different percolation
transitions under product rule are actually continuous. In Ref. Costa10 ,
Costa et al. Costa10  studied a model with stronger suppression of large
clusters and its percolation transition is continuous. Lee et al. Lee11
concluded that the explosive percolation is continuous by studying cluster
size distribution. Grassberger et al. Grassberger11  found that the explosive
transition is still continuous but with unusual finite-size behavior. In the
works of Liu et al. Liu11  and Fan et al. Fan12 , a generalized Achlioptas
process (GAP) was introduced to investigate the percolation transitions of
two-dimensional lattice network and Erdös-Rényi (ER) network ER  respectively.
In the generalized Achlioptas process, the edge with minimum product of
cluster sizes is connected with a probability $p$ from two randomly picked
unoccupied edges. When $p=1/2$, GAP recovers to a random process. With $p=1$,
GAP becomes AP. The percolation transitions of two-dimensional lattice network
and ER network under GAP keep to be continuous. Their critical exponents and
and therefore universality classes depend on the probability parameter $p$.
With a rigorous mathematical proof, Riordan and Warnke Riordan11  concluded
that the percolation transitions under all Achlioptas processes are
continuous.

The site percolation on a two-dimensional square lattice under product rule
was first investigated by Choi et al. Choi11 . They claimed that the
percolation is discontinuous from the non-vanishing hysteresis between the
directed and reverse process for the fraction of the sites belonging to the
largest cluster. However, Bastas et al. Bastas11  demonstrated that the
hysteresis phenomena will disappear in the thermodynamic limit. Using the
finite-size behaviors of the largest cluster and its standard deviation, they
concluded that the explosive percolation transition is continuous but belongs
to different universality class. To clarify this controversy, further
investigations are needed.

The notation of a universality class is a basic tenet in the physics of
critical phenomena. Within a universality class, the universal quantities such
as critical exponents and scaling functions are independent of microscopic
details. It is accepted traditionally that a universality class is
characterized by the spatial dimensionality of the system and by the number of
the components of the order parameter. (See, e.g., the review article PAH91 .)
On a two-dimensional lattice, the random bond percolation has the same
critical exponents as the random site percolation. The random bond and site
percolation belong to the same universality class. For the bond percolation
under GAP with $p>1/2$, their critical exponents are different from that of
random bond percolation and depend on the probability parameter $p$ Liu11 .
The universality classes of bond percolation under GAP are different from that
of random bond percolation Liu11 . It is of interest to investigate the
critical behaviors and universality class of site percolation in relation to
the Achlioptas process.

In this paper, we study the critical behaviors of site percolation under AP in
two-dimensional square lattice. We investigate the sizes of giant clusters in
the lattice. From the finite-size scaling behaviors of giant clusters, we can
determine the percolation transition point and the corresponding critical
exponents. Both the critical exponents of the sizes of giant clusters and
their size gaps during evolution Lee11 ; Nagler11 ; Fan13  are calculated.

Our paper is organized as following. In Section 2, we present the finite-size
scaling behaviors of giant clusters near the phase transition point of
percolation. The finite-size scaling behaviors of giant clusters for site
percolation under random and product rules on two-dimensional lattice are
studied in Sections 3 and 4 respectively. The universality classes of bond and
site percolation under random and product rules are discussed in Section 5.

##  II Critical behaviors of percolation phase transition

The percolation phase transition in a network with $N=L^{d}$ nodes is
indicated by the appearance of a giant cluster whose size becomes comparable
with $N$. The size $S_{1}$ of the giant cluster is taken as the order-
parameter of percolation phase transition. Near a critical point, the size
$S_{2}$ of the second largest cluster in the network becomes comparable with
$N$ also. Both $S_{1}$ and $S_{2}$ demonstrate critical behavior near the
critical point Liu11 ; Fan12 .

Following the finite-size scaling of order-parameter privman1 ; privman2 , we
anticipate that the reduced sizes of $S_{1}$ and $S_{2}$ in a network with $N$
nodes and $N_{r}$ edges follow the finite-size scaling form as Liu11 ; Fan12

$\displaystyle s_{1}(r,L)\equiv S_{1}/L^{d}$ $\displaystyle=$ $\displaystyle
L^{-\beta/\nu}\,\tilde{s}_{1}(tL^{1/\nu}),$ (1) $\displaystyle
s_{2}(r,L)\equiv S_{2}/L^{d}$ $\displaystyle=$ $\displaystyle
L^{-\beta/\nu}\,\tilde{s}_{2}(tL^{1/\nu}),$ (2)

where $r=N_{r}/N$ is the reduced number of edges and $t=(r-r_{c})/r_{c}$ is
the deviation from critical point $r_{c}$. The scaling variable $tL^{1/\nu}$
is related to the size ratio $L/\xi$ of $L$ to the correlation length in bulk
$\xi=\xi_{0}|t|^{-\nu}$. The finite-size scaling form is supposed to be valid
in the asymptotic critical region with $L\gg 1$ and $|t|\ll 1$.

From Eqs.(1) and (2), the finite-size scaling form of size ratio $s_{2}/s_{1}$
is obtained as

$s_{2}/s_{1}=\tilde{s}_{2}(tL^{1/\nu})/\tilde{s}_{1}(tL^{1/\nu})\equiv
U(tL^{1/\nu}).$ (3)

At the critical point $r_{c}$, the size ratio $s_{2}/s_{1}=U(0)$ and becomes
independent of system size $L$. The curves $s_{2}/s_{1}$ of different $L$
against $r$ have a cross-point, which can be used to determine the critical
point $r_{c}$.

The logarithm of Eq.(1) is

$\ln s_{1}(r,L)=-(\beta/\nu)\ln L+\ln\tilde{s}_{1}(tL^{1/\nu}).$ (4)

At the critical point, we have

$\ln s_{1}(r_{c},L)=-(\beta/\nu)\ln L+\ln\tilde{s}_{1}(0)\;,$ (5)

which is a straight line against $\ln L$. We can fix the critical point
$r_{c}$ by the straight line of $\ln s_{1}(r,L)$ with respect to $\ln L$. From
the slope of the straight line, we can determine the critical exponent ratio
$\beta/\nu$. By introducing scaling variable $tL^{1/\nu}$ to make curves
$s_{2}/s_{1}$ of different $L$ collapse into a scaling function
$U(tL^{1/\nu})$, the critical exponent $\nu$ can be estimated.

There are critical behaviors also during the evolution of a network. The
percolation phase transition is accompanied not only by the appearance of a
giant cluster but also by the giant size jump of the largest cluster. In the
Refs.Nagler11  and Fan13 , the percolation phase transition was investigated
by the size jumps of the largest cluster.

In a simulation of network evolution, edges are added under some rule one by
one into network. At an evolution step $T$, the number of edges in the network
increases from $T-1$ to $T$ and the size of the largest cluster varies
correspondingly from $S_{1}(T-1)$ to $S_{1}(T)$. At this step, the largest
cluster has a reduced size jump
$\delta_{T}\equiv\left[S_{1}(T)-S_{1}(T-1)\right]/N$. The largest reduced size
jump of the whole evolution process is

$\Delta\equiv max\left\\{\delta_{1},\delta_{2},\delta_{3},...\right\\}.$ (6)

The evolution step, where the largest size jump appears, is denoted as
$T_{c}$. In the $i$-th simulation, we obtain the largest reduced size jump
$\Delta^{(i)}$ and the critical reduced evolution step
$r_{c}^{(i)}=T_{c}^{(i)}/N$. From the results of $M$ simulations, we can
calculate following averages

$\displaystyle\bar{\Delta}(L)$ $\displaystyle=$
$\displaystyle\frac{1}{M}\sum_{i=1}^{M}\Delta^{(i)}\;,$ (7)
$\displaystyle\bar{r}_{c}(L)$ $\displaystyle=$
$\displaystyle\frac{1}{M}\sum_{i=1}^{M}r_{c}^{(i)}\;.$ (8)

In the bulk limit $L\to\infty$, we suppose that $\bar{r}_{c}(L)$ approaches
its bulk limit $r_{c}(\infty)$ as

$\bar{r}_{c}(L)=r_{c}(\infty)+a_{r}L^{-1/\nu_{1}}.$ (9)

The bulk limit $r_{c}(\infty)$ from the network evolution should be equal to
the critical point $r_{c}$ obtained from the largest cluster.

The character of percolation phase transition can be determined from the
finite-size effect of $\bar{\Delta}(L)$. If $\bar{\Delta}(L)$ approaches a
non-zero value in the limit $L\to\infty$, the percolation is a discontinuous
phase transition. For a continuous percolation phase transition,
$\bar{\Delta}(L)$ has a power-law finite-size effect as

$\bar{\Delta}(L)=a_{\Delta}L^{-\beta_{1}}.$ (10)

In the $i$-th simulation of network evolution, there are fluctuations $\delta
r_{c}=r_{c}^{(i)}-\bar{r}_{c}(L)$ and
$\delta\Delta=\Delta^{(i)}-\bar{\Delta}(L)$. Their root mean squares are
calculated as

$\displaystyle\chi_{r}$ $\displaystyle\equiv$ $\displaystyle\sqrt{<(\delta
r_{c})^{2}>},$ (11) $\displaystyle\chi_{\Delta}$ $\displaystyle\equiv$
$\displaystyle\sqrt{<(\delta\Delta)^{2}>}.$ (12)

Their dependence on network size $L$ is described by the exponents $\nu_{2}$
and $\beta_{2}$ as

$\displaystyle\chi_{r}$ $\displaystyle\propto$ $\displaystyle L^{-1/\nu_{2}},$
(13) $\displaystyle\chi_{\Delta}$ $\displaystyle\propto$ $\displaystyle
L^{-\beta_{2}}.$ (14)

In Ref. Nagler11 , the asymptotic behavior of $\bar{\Delta}(L)$ was used to
judge the continuity of a percolation phase transition. The so-called upper
pseudo transition point in Ref. Lee11  was estimated from $\bar{r}_{c}(L)$.
The general clique percolation phase transition in random networks are
identified by studying the averages and fluctuations of $\Delta$ and $r_{c}$
Fan13 .

We will investigate the above critical behaviors of a percolation phase
transition, which are characterized by the critical exponents $\beta$,
$\beta_{1}$,$\beta_{2}$, $\nu$,$\nu_{1}$ and $\nu_{2}$. The different
percolation phase transitions with the same critical exponents belong to the
same universality class.

##  III Random site percolation on two-dimensional square lattice

On a two-dimensional square lattice, the percolation transition can be
investigated both for site and bond. For the site percolation, we consider a
$L\times L$ square lattices with periodic boundary conditions in both
directions. At first, there are $N=L\times L$ unoccupied sites in a lattice.
Then the sites in the lattice are occupied one by one under some rule.
Clusters consisting of adjacent sites appear during the evolution. When
$N_{r}$ sites are occupied in the lattice, we define a reduced number of
occupied sites $r=N_{r}/N$. If the size of the largest cluster becomes
comparable with $N$, there is a site percolation in the lattice.

For the random site percolation (SP), sites are added randomly. It is known
that SP on two-dimensional square lattice occurs at $r_{c}=0.59276421(13)$
Newmann00 . From Coulomb gas arguments Nienhuis  and conformal field theory
Cardy , the critical exponents $\beta=5/36$ and $\nu=4/3$ were predicted.

In our Monte Carlo simulation of site percolation, the algorithm of Newmann
and Ziff Newmann00 ; Newmann01  has been used. We have taken lattice sizes
$L=32,64,128,256,512$,$1024$,$2048$,$4096$ and $8192$ in our simulations.
$12,800,000$ independent simulations have been run for each lattice size.

Figure 1: Size ratio of the second largest cluster to the largest cluster
$U=s_{2}/s_{1}$. The critical point of site percolation $r_{c}=0.59276(3)$ is
obtained from its cross-point.

In Fig.1, the size ratio $s_{2}/s_{1}$ is plotted with respect to $r$ for
three different lattice sizes. From its cross-point, we get the critical point
$r_{c}=0.59276(3)$ which agrees with the results of Ref.Newmann00 .

In Fig.2, $\ln s_{1}$ is shown with respect to $\ln L$ for three different
$r$. At $r=0.5927$, the curve is nearly a straight line with slope equal to
$-0.108$. The curvature becomes negative for $r=0.5924$ and positive for
$r=0.5930$. We can conclude that the critical point $r_{c}=0.5927(3)$ and the
critical exponent ratio $\beta/\nu=0.108(4)$, which agrees well with the exact
value $\beta/\nu=5/48$. Within error bounds, the critical point from $\ln
s_{1}$ agrees with that from $s_{2}/s_{1}$. Using $1/\nu=0.75$ in Fig.3, three
curves of $s_{2}/s_{1}$ at $L=256,512,1024$ collapse into a scaling function
function $U(tL^{1/\nu})$.

Figure 2: Log-log plot of $s_{1}$ with respect to $L$ for three different
reduced number of occupied sites $r$. At $r_{c}=0.5927(3)$, the curve becomes
a straight line and its slope gives the critical exponent ratio
$\beta/\nu=0.108$.

Figure 3: Finite-size scaling function $U(tL^{1/\nu})$ of the size ratio
$s_{2}/s_{1}$ for random site percolation with $1/\nu=0.75$.

In Fig.4, we show critical behaviors of the largest cluster during the
evolution of SP. The Monte Carlo simulation results of $\bar{r}_{c}(L)$ are
demonstrated in Fig.4(A). Using $r_{c}(\infty)=0.5927(3)$ from the above
investigation, a log-log plot of $r_{c}(\infty)-\bar{r}_{c}(L)$ with respect
to $L$ is made. Our Monte Carlo data demonstrate a linear dependence of
$\ln[r_{c}(\infty)-\bar{r}_{c}(L)]$ on $\ln L$. From its slope, we obtain the
inverse of exponent $1/\nu_{1}=0.75(1)$. In Fig.4(B), the power-law finite-
size behavior of $\bar{\Delta}(L)$ in Eq.(10) is confirmed by the Monte Carlo
simulation data. The exponent $\beta_{1}=0.104(1)$ is calculated from the
slope of the straight line.

The root mean squares of fluctuations $\chi_{r}$ and $\chi_{\Delta}$ are shown
in Figs.4(C) and (D). Our Monte Carlo simulation data of $\chi_{r}$ follow a
power-law dependence on $L$ with the exponent $1/\nu_{2}=0.74(1)$. The power-
law behavior of $\chi_{\Delta}$ in Eq.14 is confirm by our simulation data and
we obtain $\beta_{2}=0.104(1)$.

We have investigated also the evolution critical behaviors of random bond
percolation (BP) on two-dimensional square lattice. These results of BP are
given in Table.1. We can see that the critical exponents of SP are equal to
that of BP. Further, the critical exponent ratio $\beta/\nu$ is equal to
$\beta_{1}$ and $\beta_{2}$ and $\nu$ is equal to $\nu_{1}$ and $\nu_{2}$
within error bounds.

Figure 4: Critical behaviors of random site percolation during network
evolution: (A)log-log plot of $r_{c}(\infty)-\bar{r}_{c}(L)$ with
$r_{c}(\infty)=0.5927$ and the exponent $1/\nu_{1}=0.75(1)$; (B)log-log plot
of $\bar{\Delta}(L)$ with the exponent $\beta_{2}=0.104(1)$; (C)log-log plot
of $\chi_{r}$ with the exponent $1/\nu_{2}=0.74(1)$; (D)log-log plot of
$\chi_{\Delta}$ with the exponent $\beta_{2}=0.104(1)$.

##  IV Site percolation under product rule on two-dimensional square lattice

In this section, we consider the site percolation under product rule (SPPR).
We take the product rule Choi11 ; Bastas11  for the site evolution as follows:
(1)selecting two unoccupied sites randomly; (2)calculating the product of
sizes of the clusters which are connected by two chosen sites respectively;
(3)the site with smaller product is occupied.

The reduced sizes $s_{1}$ of the largest cluster for lattice size $L$ from
$512$ to $8192$ are shown in Fig.5. $s_{1}$ is nearly zero at small $r$ and
becomes finite when $r$ is larger than a critical value $r_{c}$. Finite
$s_{1}$ indicates the emergence of a giant cluster and a site percolation
transition.

Figure 5: Reduced size of the largest cluster as a function of $r$ at lattice
size $L=512,1024,2048,4096,8192$.

Figure 6: Size ratio of the second largest cluster to the largest cluster
$U=s_{2}/s_{1}$ at different lattice sizes. Its fixed point gives the critical
point $r_{c}=0.7634(2)$.

Figure 7: $\ln s_{1}$ with respect to $\ln L$ around the critical point
$r_{c}=0.7634(5)$, where the curve becomes a straight line with slope equal to
$-0.010$.

Figure 8: Finite-size scaling function $U(tL^{1/\nu})$ of size ratio
$s_{2}/s_{1}$ with $1/\nu=0.42$.

The results obtained from the size of the largest cluster are shown in Figs.
6, 7 and 8. From the fixed point of $s_{2}/s_{1}$ in Fig.6, we can get the
critical point $r_{c}=0.7634(2)$. The straight line in Fig.7 gives the
critical point $r_{c}=0.7634(5)$, which is in agreement with the value of the
fixed point. The accuracy of our critical point is higher than that of
$r_{c}=0.768(3)$ in Ref.Choi11  and $r_{c}=0.756(6)$ in Ref.Bastas11 .

The slope of the straight line in Fig.7 gives the critical exponent ratio
$\beta/\nu=0.010(1)$, which is different from $\beta/\nu=0.04\pm 0.02$ of
Ref.Bastas11 .

We can see in Fig.8 that the size ratios $s_{2}/s_{1}$ of different lattice
sizes collapse well with the scaling variable $tL^{1/\nu}$ for data of $L\geq
512$. This means that the asymptotic region of SPPR is more restricted than
that of SP. In the finite-size scaling plot of $s_{2}/s_{1}$, we use the
inverse of the critical exponent $1/\nu=0.42(1)$.

The size ratio at critical point $U(0)$ is supposed to be universal. We obtain
$U(0)=0.03$ for SPPR and $U(0)=0.1$ for SP. This difference of $U(0)$
indicates also that SPPR and SP belong to different universality class.

We have studied also the size jumps of the largest cluster during network
evolution for SPPR. The different quantities related to the size jump are
shown in Fig.9 with respect to the lattice size $L$. From Fig.9(A), we can
conclude that the average transition point approaches to its bulk limit
according the power-law in Eq.11. Using the Monte Carlo data of the lattice
size $l\geq 512$, we get the inverse of exponent $1/\nu_{1}=0.42(2)$. The root
mean square of the fluctuations of transition point is plotted in Fig.9(C) and
a power-law in Eq.13 is found $1/\nu_{2}=0.42(3)$. The simulation results of
the size jumps of the largest cluster are presented in Figs.9(B) and (D). The
average size jump $\bar{\Delta}$ shows a quite different $L$-dependence from
that of the random site percolation. No power-law behavior is found for
$\bar{\Delta}(L)$, which increases with $L$ at first and then decreases. The
root mean square of fluctuations of $\Delta$ follows a power-law of the
lattice size also. From Monte Carlo simulation data, we get
$\beta_{2}=0.008(4)$.

For comparison, we have studied also the evolution critical behaviors of bond
percolation under product rule (BPPR). The critical exponents related to the
cluster size jump $\beta_{1}=0.044(5)$ and $\beta_{2}=0.041(4)$ are obtained.
For the exponents related to critical point, we get $1/\nu_{1}=0.99(5)$ and
$1/\nu_{2}=0.94(3)$. The critical exponent ratio $\beta/\nu$ and the inverse
of critical exponent $1/\nu$ of BPPR have been calculated in Ref.Liu11 .
$\beta/\nu=0.064(3)$ is different from the exponents $\beta_{1}$ and
$\beta_{2}$. Within error bounds, $1/\nu=0.93(1)$ is equal to $1/\nu_{1}$ and
$1/\nu_{2}$. All results of BP, SP, BPPR and SPPR are summarized in Table.1.
We can see that the critical exponents of BPPR are different from that of
SPPR. The universality classes of BPPR and SPPR are different.

Figure 9: Results of site percolation under the product rule: (A)log-log plot of $r_{c}(\infty)-\bar{r}_{c}(L)$ with $r_{c}(\infty)=0.7634$ and the exponent $1/\nu_{1}=0.42(2)$; (B)log-log plot of $\bar{\Delta}(L)$, which increases at first and then decreases slowly; (C)log-log plot of $\chi_{r}$ with the exponent $1/\nu_{2}=0.42(3)$; (D)log-log plot of $\chi_{\Delta}$ with the exponent $\beta_{2}=0.008(4)$.  Table 1: Summary of critical points and critical exponents of BP, SP, BPPR and SPPR. $r_{c}$, $\beta/\nu$ and $1/\nu$ of BPPR are taken from Ref.Liu11 . |  BP |  SP |  BPPR |  SPPR
---|---|---|---|---
$r_{c}$ |  0.5000(4) |  0.5927(3) |  0.5266(1) |  0.7634(2)
$1/\nu$ |  0.75 |  0.75(1) |  0.93(1) |  0.42(1)
$1/\nu_{1}$ |  0.75(1) |  0.75(1) |  0.99(5) |  0.42(2)
$1/\nu_{2}$ |  0.74(1) |  0.74(1) |  0.94(3) |  0.42(3)
$\beta/\nu$ |  0.108(4) |  0.108(4) |  0.064(3) |  0.010(1)
$\beta_{1}$ |  0.105(2) |  0.104(1) |  0.044(5) |
$\beta_{2}$ |  0.104(1) |  0.104(1) |  0.041(4) |  0.008(4)

##  V Conclusion

We have studied the critical behaviors of the sizes and the size jumps of
giant clusters on two-dimensional square lattice for both site and bond
percolation under random and product rules respectively. Under the random
rule, sites or bonds are added randomly into the lattice. We add the site or
bond with smaller size product of two connected clusters into the lattice from
two candidates picked randomly when the product rule is taken. The finite-size
scaling forms of giant cluster size and size jumps are proposed and used to
determine the transition points and critical exponents of the percolation
transitions from Monte Carlo data.

Our results show that the critical exponents of the size and size jump of
random site percolation are equal to that of random bond percolation within
error bounds of Monte Carlo data. As expected, the universality class of
percolation is independent of site and bond under the random rule. The
critical exponents of the size and size jump in the site percolation under the
product rule are different from that in the random percolation and also the
bond percolation under the product rule. The site percolation and bond
percolation under the product rule do not belong to the same universality
class.

This work is supported by the National Natural Science Foundation of China
under grant 11121403.

## References

  * (1) D. Achlioptas, R. M. D’Sousa and J. Spencer, Science 323, 1453 (2009).
  * (2) R. M. Ziff, Phys. Rev. Lett. 103, 045701 (2009).
  * (3) R. M. Ziff, Phys. Rev. E. 82, 051105 (2010).
  * (4) F. Radicchi and S. Fortunato, Phys. Rev. E. 81, 036110 (2010).
  * (5) Y. S. Cho, J. S. Kim, J. Park, B. Kahng, and D. Kim, Phys. Rev. Lett. 103, 135702 (2009).
  * (6) F. Radicchi and S. Fortunato, Phys. Rev. Lett. 103, 168701 (2009).
  * (7) R. A. da Costa, S. N. Dorogovtsev, A. V. Goltsev, and J. F. F. Mendes, Phys. Rev. Lett. 105, 255701 (2010).
  * (8) H. K. Lee, B. J. Kim, and H. Park, Phys. Rev. E. 84, 020101(R) (2011).
  * (9) P. Grassberger, C. Christensen, G. Bizhani, S. W. Son, and M. Paczuski, Phys. Rev. Lett. 106, 225701 (2011).
  * (10) M.X. Liu, J.F. Fan, L.S. Li, and X.S. Chen, Eur. Phys. J. B 85, 132 (2012).
  * (11) J.F. Fan, M.X. Liu, L.S. Li, and X.S. Chen, Phys. Rev. E 85, 061110 (2012).
  * (12) P. Erdös and A. Rényi, Publ. Math. Inst. Hungar. Acad. Sci.5, 17 (1960).
  * (13) O. Riordan and L. Warnke, Science 333, 322 (2011).
  * (14) W. Choi, S. H. Yook, and Y. Kim, Phys. Rev. E. 84, 020102(R) (2011).
  * (15) N. Bastas, K. Kosmidis, and P. Augyrakis, Phys. Rev. E. 84, 066112 (2011).
  * (16) V. Privman, A. Aharony, and P.C. Hohenberg, in Phase Transition and Critical Phenomena, edited by C. Domb and J.L. Lebowitz (Academic, New York, 1991), Vol. 14, P.1.
  * (17) J. Nagler, A. Levina and M. Timme, Nat. Phys. 7, 265 (2011).
  * (18) J.F. Fan and X.S. Chen, Europhys. Lett. 107, 28005 (2014).
  * (19) M. E. J. Newmann and R. M. Ziff, Phys. Rev. Lett. 85, 4104 (2000).
  * (20) B. Nienhuis, in Phase Transition and Critical Phenomena, edited by C. Domb and J. L. Lebowitz, Vol. 11 (Academic, London, 1987).
  * (21) J. L. Cardy, in Phase Transition and Critical Phenomena, edited by C. Domb and J. L. Lebowitz, Vol. 11 (Academic, London, 1987).
  * (22) M. E. J. Newmann and R. M. Ziff, Phys. Rev. E. 64, 016706 (2001).
  * (23) V. Privman and M. E. Fisher, Phys. Rev. B. 30, 322 (1984).
  * (24) V. Privman, Finite-size Scaling and Numerical Simulation of Statistical Systems, (World Scientific, Singapore, 1900).