Explosive percolation on the Bethe lattice is ordinary

Abstract

The Achlioptas process, which suppresses the aggregation of large-sized clusters, can exhibit an explosive percolation (EP) where the order parameter emerges abruptly yet continuously in the thermodynamic limit. It is known that EP is accompanied by an abnormally small critical exponent of the order parameter. In this paper, we report that a novel type of EP occurs on a Bethe lattice, where the critical exponent of the order parameter is the same as in ordinary bond percolation based on numerical analysis. This is likely due to the property of a finite Bethe lattice that the number of sites on the surface with only one neighbor is extensive to the system size. To overcome this finite size effect, we consider an approximate size of the cluster that each site on the surface along its branch belongs to, and accordingly approximate the sizes of an extensive number of clusters during simulation. As a result, the Achlioptas process becomes ineffective and the order parameter behaves like that of ordinary percolation at the threshold. We support this result by measuring other critical exponents as well.

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Appendix A: Observation of the same critical exponent values in the original model

In this Appendix, we show that $P_{\infty } \propto (p-p_c)^{\beta }$ as $p \rightarrow p^+_c$ and $\chi \propto |p-p_c|^{-\gamma (\gamma ')}$ as $p \rightarrow p_c^+ (p_c^-)$, with $\beta \approx 1$ and $\gamma \approx \gamma ' \approx 1$ reproduced in the original EP model on the Bethe lattice introduced in [31]. Here, we do not investigate the $n_s$ of this model for simplicity.

We begin with the self-consistent equations introduced in the previous study. The previous study assumed that $P_{\infty }$ and $\chi$ obtained using $P_{n'st}$ of an internal Cayley tree with radius $n'$ for any $n' \le n$ should be the same as those obtained using $P_{nst}$. Based on this assumption, two self-consistent equations were derived as follows.

To introduce the first equation, we denote $P_{\infty }$ obtained using an internal Cayley tree with the radius $n'$ as $P_{n' \infty }$, where it is given by $P_{n' \infty } = 1-\sum _{s,t}P_{n' st}(1-A)^t$. Then the above assumption implies that $P_{n\infty }=P_{n' \infty }=P_{\infty }$ for any $n' \le n$, which leads to the first equation

$$\begin{aligned} \sum _{s,t}P_{nst}(1-A)^t = \sum _{s,t}P_{n'st}(1-A)^t \end{aligned}$$

(A1)

for an arbitrary $n'< n$.

To introduce the second equation, we denote $\chi$ obtained using an internal Cayley tree with radius $n'$ as $\chi _{n'}$, where $\chi _{n'} = (1-P_{\infty })^{-1}\sum _{s,t}(s-t+tS_b)P_{n' st}(1-A)^t$. Then the above assumption implies that $\chi _{n}=\chi _{n'}=\chi$ for any $n' \le n$, which leads to the second equation

$$\begin{aligned}&\sum _{s,t}(s-t+tS_b)P_{nst}(1-A)^t \nonumber \\&=\sum _{s,t}(s-t+tS_b)P_{n'st}(1-A)^t \end{aligned}$$

(A2)

for an arbitrary $n'<n$.

In principle, Eqs. (A1) and (A2) for a common $n' < n$ can be solved by iterating until $A^{(i)}$ and $S_b^{(i)}$ converge to A and $S_b$, thereby satisfying these equations following the process described in [31]. Then $P_{\infty }$ and $\chi$ are obtained by substituting A and $S_b$ into Eq. (2) and (5).

Fig. 4

Simulation result of the original EP model with $(n, n')=(10, 4)$ ($\bullet$) and $(n, n')=(10, 1)$ ($\blacksquare$). a $S_b$ for $(n,n')=(10,4)$ becomes nonzero as p exceeds the dotted line. b $S_b$ in both conditions differs over the entire range of p

Fig. 5

Simulation result of the original EP model on the Bethe lattice with $z=4$. a $P_{\infty }(p)$ for $(n,n')=(11, 4)$. Inset: Jump size of $P_{\infty }$ at $p=p_c$ vs. N for fixed $n'=4$. (b–d) Data obtained using $n=8$ ($\square$), 10 ($\circ$), 11 ($\triangle$), and 13 ($\bullet$) for fixed $n'=4$. The slopes of the dotted lines are b 1, c $-1$, and d $-1$

However, we find that $S_b$ depends on $n'$ for a fixed n, as shown in Fig. 4, which violates the assumption that $S_b$ is the same irrespective of $n'$ for a fixed n. We can understand this as follows. At first, we consider a small p range where $A=0$ and $P_{nst}=0$ for all $t \ge 1$. In this case, Eq. (A2) is reduced to $\sum _{s}sP_{ns0} = \sum _{s,t}(s-t+tS_b)P_{n'st}$. Accordingly, there must be at least one $t>0$ satisfying $P_{n'st}>0$ in order for $S_b > 1$. As $n'$ becomes larger, the average size of the cluster to which O belongs must be larger to satisfy this condition. Therefore, the initial range of p where $S_b=1$ increases with $n'$, as shown in Fig. 4a. This initial difference between $S_b$ for different $n'$ leads to a difference between those in the entire range of p, as shown in Fig. 4b. Note that in the main text we suggested our method to avoid this dependence of $S_b$ on $n'$.

In Fig. 5, we show that the critical exponent values $\beta \approx 1$ and $\gamma \approx \gamma ' \approx 1$ are also observed in the original EP model for the values of $(n, n')$ used in [31]. As mentioned in [31], the average gap size $\langle P^{(i)}_{\infty }(p^{(i)}_c) \rangle > 0$ for a finite n and $\langle P^{(i)}_{\infty }(p^{(i)}_c)\rangle \rightarrow 0$ as $n \rightarrow \infty$ (Fig. 5a). To estimate $\beta$ in the thermodynamic limit $n \rightarrow \infty$, we measure the slope of $\langle P^{(i)}_{\infty }(p) \rangle - \langle P^{(i)}_{\infty }(p^{(i)}_c)\rangle$ with respect to $(p-p_c)$, where $p_c$ satisfies $\langle P^{(i)}_{\infty }(p_c) \rangle = \langle P^{(i)}_{\infty }(p^{(i)}_c)\rangle$. As shown in Fig. 5b, the range of $p-p_c$ where the slope is close to 1 increases with n, which supports $\beta \approx 1$. Finally, $\gamma \approx 1$ and $\gamma ' \approx 1$ are shown by plotting $\langle \chi (p_c^{(i)}+\overline{p}) \rangle$ with respect to $\overline{p}$ and $-\overline{p}$ in Fig. 5c, d, respectively.