Continuity of the Explosive Percolation Transition

TL;DR

This study uses extensive simulations to analyze explosive percolation on the complete graph, revealing that the transition is continuous rather than discontinuous, with a well-characterized power-law cluster distribution.

Contribution

The paper provides evidence that the explosive percolation transition is continuous, challenging previous beliefs of its discontinuity, through finite-size scaling analysis and distribution collapse.

Findings

Cluster-size distribution follows a power-law with exponent ~2.06

Distribution collapse onto a single scaling curve for large N

Transition converges to a well-defined percolation threshold

Abstract

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 $\tau = 2.06(2)$, 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 $N\rightarrow\infty$. 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.