Explosive Percolation: Unusual Transitions of a Simple Model

TL;DR

This paper reviews recent advances in explosive percolation, a sharply transitioning process in network theory, highlighting its initial perceived discontinuity, later understanding as continuous with unusual scaling, and outlining future research directions.

Contribution

It provides a comprehensive review of explosive percolation, clarifies its transition nature, and discusses open problems and future research directions.

Findings

Transition initially thought discontinuous

Later shown to be continuous with unusual scaling

Highlights open problems in the field

Abstract

In this paper we review the recent advances on explosive percolation, a very sharp phase transition first observed by Achlioptas et al. (Science, 2009). There a simple model was proposed, which changed slightly the classical percolation process so that the emergence of the spanning cluster is delayed. This slight modification turns out to have a great impact on the percolation phase transition. The resulting transition is so sharp that it was termed explosive, and it was at first considered to be discontinuous. This surprising fact stimulated considerable interest in "Achlioptas processes". Later work, however, showed that the transition is continuous (at least for Achlioptas processes on Erdos networks), but with very unusual finite size scaling. We present a review of the field, indicate open "problems" and propose directions for future research.