Continuity of the explosive percolation transition

The explosive percolation problem on the complete graph is investigated via extensive numerical simulations. We obtain the cluster-size distribution at the moment when the cluster size heterogeneity becomes maximum. The distribution is found to be well described by the power-law form with the decay exponent $\tau =2.06\left(2\right)$, followed by a hump. We then use the finite-size scaling method to make all the distributions at various system sizes up to $N=2^{37}$ collapse perfectly onto a scaling curve characterized solely by the single exponent $\tau$. We also observe that the instant of that collapse converges to a well-defined percolation threshold from below as $N\to \infty$. Based on these observations, we show that the explosive percolation transition in the model should be continuous, contrary to the widely spread belief of its discontinuity.

Figure 1

Convergence of $t_l$ (the lower branch) and $t_u$ (the upper one) averaged for $100\sim 5000$ different realizations for each system size ($N=2^{17},2^{18},\cdots ,2^{37}$). Errors are smaller than symbol sizes, if not shown explicitly. Lines are just guides to the eyes. As $N$ increases, $t_u$ ($t_l$) decreases (increases), approaching to the well-defined value in the thermodynamic limit displayed as the horizontal line. For clarity, the upper branch is vertically enlarged in the inset.

Figure 2

Evolutions of the largest cluster size (the upper curve) and the size of the second largest cluster (the lower curve) versus the growth step $t$. As $t$ crosses $t_u$ from below, the size of the largest cluster exhibits a sudden biggest increase since the maximum second-largest cluster merges into it at that moment. The system size is $N=2^{23}$.

Figure 3

Log-binned cluster-size distributions at $t_u$ for each $N$, where the horizontal axis is the cluster size $s$ divided by $N$ for convenience. The solid line above is a guiding line of which the decay exponent is 2.06.

Figure 4

Log-binned cluster-size distributions at $t_l$ for each $N$. The decay exponent of the guiding solid line below is 2.06.

Figure 5

Finite-size scaling of the cluster size distribution $n\left(s\right)$ at $t_l\left(N\right)$: All data points plotted in the form of $s^{\tau }n\left(s\right)$ versus $s{N}^{-1/\tau }$ at $\tau =2.06$ collapse into a single curve. (Inset) The same plot but with $\tau =2$. This clearly shows that $\tau \approx 2.06$ ($>$2) gives us the much better quality of the FSS collapse.