Local cluster aggregation models of explosive percolation

TL;DR

This paper presents simple local graph evolution models that exhibit discontinuous percolation transitions, analyzed through differential equations, revealing both abrupt giant component emergence and continuous scaling behaviors.

Contribution

The paper introduces the simplest local choice models of graph evolution that demonstrate discontinuous percolation transitions and are analytically tractable via differential equations.

Findings

Models exhibit discontinuous giant component emergence.

Scaling exponents differ from classic Erdos-Renyi model.

Evolution accurately described by differential equations.

Abstract

We introduce perhaps the simplest models of graph evolution with choice that demonstrate discontinuous percolation transitions and can be analyzed via mathematical evolution equations. These models are local, in the sense that at each step of the process one edge is selected from a small set of potential edges sharing common vertices and added to the graph. We show that the evolution can be accurately described by a system of differential equations and that such models exhibit the discontinuous emergence of the giant component. Yet, they also obey scaling behaviors characteristic of continuous transitions, with scaling exponents that differ from the classic Erdos-Renyi model.