Watersheds and Explosive percolation

TL;DR

This paper reviews models of explosive percolation, showing that controlling the largest cluster size leads to discontinuous transitions with fractal cluster surfaces.

Contribution

It demonstrates that suppressing large deviations in cluster size is key to achieving discontinuous percolation transitions.

Findings

Discontinuous transition occurs when largest cluster growth is controlled.

Gaussian cluster-size distribution and compact clusters are observed.

Cluster surfaces are fractal with the same dimension as watershed lines.

Abstract

The recent work by Achlioptas, D'Souza, and Spencer opened up the possibility of obtaining a discontinuous (explosive) percolation transition by changing the stochastic rule of bond occupation. Despite the active research on this subject, several questions still remain open about the leading mechanism and the properties of the system. We review the largest cluster and the Gaussian models recently introduced. We show that, to obtain a discontinuous transition it is solely necessary to control the size of the largest cluster, suppressing the growth of a cluster differing significantly, in size, from the average one. As expected for a discontinuous transition, a Gaussian cluster-size distribution and compact clusters are obtained. The surface of the clusters is fractal, with the same fractal dimension of the watershed line.