Explosive percolation on a scale-free multifractal weighted planar stochastic lattice

In this article, we investigate explosive bond percolation (EBP) with the product rule, formally known as the Achlioptas process, on a scale-free multifractal weighted planar stochastic lattice. One of the key features of the EBP transition is the delay, compared to the corresponding random bond percolation (RBP), in the onset of the spanning cluster. However, when it happens, it happens so dramatically that initially it was believed, although ultimately proved wrong, that explosive percolation (EP) exhibits a first-order transition. In the case of EP, much effort has been devoted to resolving the issue of its order of transition and almost no effort has been devoted to finding the critical point, critical exponents, etc., to classify it into universality classes. This is in sharp contrast to the situation for classical random percolation. We do not even know all the exponents of EP for a regular planar lattice or for an Erdös-Renyi network. We first find the critical point $p_c$ numerically and then obtain all the critical exponents, $\beta$, $\gamma$, and $\nu$, as well as the Fisher exponent $\tau$ and the fractal dimension $d_f$ of the spanning cluster. We also compare our results for EBP with those for RBP and find that all the exponents of EBP obey the same scaling relations as do those for RBP. Our findings suggest that EBP is not special except for the fact that the exponent $\beta$ is unusually small compared to that for RBP.

Figure 1

Snapshot of the weighted planar stochastic lattice.

Figure 2

Spanning probability $W(p,L)$ vs $p$ of (a) explosive and (b) random bond percolation on WPSLs of different system sizes, $L^2=1+3t$. The spanning probability curves for different sizes cross, and at the percolation threshold we find $p_c=0.4021$ for EBP and 0.3457 for RBP. (c) We plot $log(p-p_c)$ vs $logL$ for both cases. The two lines have slopes of $1/\nu =0.8801\pm 0.0049$ and $0.6117\pm 0.0074$ for EBP and RBP, respectively. (d) We plot the dimensionless quantities $W$ vs $(p-p_c)L^{1/\nu }$ and find that the distinct plots in (a) and (b) collapse superbly into their own universal curves.

Figure 3

Percolation strength or percolation probability $P(p,L)$ of (a) explosive bond percolation (EBP) and (b) random bond percolation (RBP) on a WPSL for different system size. (c) We plot $logP$ vs $logL$ using data for a fixed value of $(p-p_c)L^{1/\nu }$ and find almost-parallel lines with slope $\beta /\nu =0.059835\pm 0.00038$ for EBP and $0.1357\pm 0.0002$ for RBP, respectively, which clearly implies that the critical exponent $\beta =0.0679$ is negligibly small for EBP compared to the $\beta =0.222$ for RBP on the same lattice. (d) We plot $P{L}^{\beta /\nu }$ vs $(p-p_c)L^{1/\nu }$ and find that the distinct plots in (a) and (b) collapse into their own scaling functions.

Figure 4

Mean cluster size $S(p,L)$ for (a) explosive and (b) random bond percolation as a function of $p$ for a WPSL of size $L$. (c) We plot $logS$ vs $logL$ using the size of $S$ for a fixed value of $(p-p_c)L^{1/\nu }$ and find almost-parallel lines with slope $\gamma /\nu$ equal to $1.8818\pm 0.0069$ and $1.7315\pm 0.0019$ for explosive and random bond percolation, respectively. (d) We plot the same data as in (a) and (b) in self-similar coordinates, namely, $S{L}^{-\gamma /\nu }$ vs $(p-p_c)L^{1/\nu }$ and find all the distinct plots in (a) and (b) collapse into their respective universal curves.

Figure 5

(a) Cluster size distribution function $log\left(n_s\left(p_c\right)\right)$ vs $logs$ for explosive and random bond percolation of a WPSL of size $L^2$ equal to 6001, 12 001, 24 001, and 36 001 and to 7501, 19 501, 40 501, and 67 603, respectively. We find almost-parallel lines with slopes of 2.030 and 2.072 8 for explosive and random bond percolation, respectively. (b) We plot $ln\left(M\right)$ vs $ln\left(L\right)$ where $M$ is the number of sites in the spanning cluster. The two lines whose slopes are $d_f=1.9415\pm 0.0055$ for EBP and slope $d_f=1.8643\pm 0.0014$ for RBP represent the fractal dimensions of their respective spanning clusters.