Criterion for explosive percolation transitions on complex networks

TL;DR

This paper establishes a criterion for explosive percolation transitions in complex networks, showing that such transitions occur when the mean cluster size diverges as the network grows, supported by simulations and analysis.

Contribution

It extends previous work by providing a general condition under which explosive percolation occurs, linking divergence of mean cluster size to the nature of the transition.

Findings

Explosive transitions occur when mean cluster size diverges before the threshold.

Networks with a non-proportional growth of clusters exhibit explosive percolation.

Simulations and analytical models support the proposed criterion.

Abstract

In a recent Letter, Friedman and Landsberg discussed the underlying mechanism of explosive phase transitions on complex networks [Phys. Rev. Lett. 103, 255701 (2009)]. This Brief Report presents a modest, though more insightful extension of their arguments. We discuss the implications of their results on the cluster-size distribution and deduce that, under general conditions, the percolation transition will be explosive if the mean number of nodes per cluster diverges in the thermodynamic limit and prior to the transition threshold. In other words, if, upon increase of the network size n the amount of clusters in the network does not grow proportionally to n, the percolation transition is explosive. Simulations and analytical calculations on various models support our findings.