Explosive Percolation in Erd\"os-R\'enyi-Like Random Graph Processes

TL;DR

This paper investigates a class of Erd"os-Rényi-like random graph processes and proves the existence of discontinuous phase transitions, contrasting with the continuous transition observed in classical Erd"os-Rényi and Achlioptas processes.

Contribution

It introduces a new class of Erd"os-Rényi-like processes and demonstrates that these processes can exhibit discontinuous phase transitions.

Findings

Discontinuous phase transitions are proven for the new class of processes.

Contrasts with classical Erd"os-Rényi process which has continuous transition.

Contrasts with Achlioptas processes which also have continuous transition.

Abstract

The evolution of the largest component has been studied intensely in a variety of random graph processes, starting in 1960 with the Erd\"os-R\'enyi process. It is well known that this process undergoes a phase transition at n/2 edges when, asymptotically almost surely, a linear-sized component appears. Moreover, this phase transition is continuous, i.e., in the limit the function f(c) denoting the fraction of vertices in the largest component in the process after cn edge insertions is continuous. A variation of the Erd\"os-R\'enyi process are the so-called Achlioptas processes in which in every step a random pair of edges is drawn, and a fixed edge-selection rule selects one of them to be included in the graph while the other is put back. Recently, Achlioptas, D'Souza and Spencer (2009) gave strong numerical evidence that a variety of edge-selection rules exhibit a discontinuous phase transition. However, Riordan and Warnke (2011) very recently showed that all Achlioptas processes have a continuous phase transition. In this work we prove discontinuous phase transitions for a class of Erd\"os-R\'enyi-like processes in which in every step we connect two vertices, one chosen randomly from all vertices, and one chosen randomly from a restricted set of vertices.