Cluster aggregation model for discontinuous percolation transition

TL;DR

This paper investigates how varying the connection kernel in Erdős-Rényi networks influences the nature of percolation transitions, revealing conditions under which these transitions become discontinuous, including from specific initial states, and elucidates the mechanisms behind explosive percolation.

Contribution

It extends the rate equation framework to sub-linear kernels, demonstrating discontinuous percolation transitions and explaining the explosive transition mechanism in Achlioptas processes.

Findings

Discontinuous PT occurs for kernels with exponent <1/2.

Discontinuous PT also arises from proper initial conditions.

The rate equation approach uncovers the mechanism of explosive PT.

Abstract

The evolution of the Erd\H{o}s-R\'enyi (ER) network by adding edges can be viewed as a cluster aggregation process. Such ER processes can be described by a rate equation for the evolution of the cluster-size distribution with the connection kernel $K_{ij}\sim ij$, where $ij$ is the product of the sizes of two merging clusters. Here, we study more general cases in which $K_{ij}$ is sub-linear as $K_{ij}\sim (ij)^{\omega}$ with $0 \le \omega < 1/2$; we find that the percolation transition (PT) is discontinuous. Moreover, PT is also discontinuous when the ER dynamics evolves from proper initial conditions. The rate equation approach for such discontinuous PTs enables us to uncover the mechanism underlying the explosive PT under the Achlioptas process.