Strongly discontinuous explosive percolation with multiple giant components

TL;DR

This paper studies a generalized explosive percolation process where multiple giant components emerge simultaneously in a strongly discontinuous transition, with the number and timing controlled by a parameter alpha.

Contribution

It introduces a generalized process controlling the emergence of multiple giant components with a strongly discontinuous transition, extending previous models.

Findings

Multiple giant components appear simultaneously during the transition.

Tuning alpha affects the delay and explosiveness of the transition.

The critical point and discontinuity are unaffected when only spanning edges are sampled.

Abstract

We generalize the random graph evolution process of Bohman, Frieze, and Wormald [T. Bohman, A. Frieze, and N. C. Wormald, Random Struct. Algorithms, 25, 432 (2004)]. Potential edges, sampled uniformly at random from the complete graph, are considered one at a time and either added to the graph or rejected provided that the fraction of accepted edges is never smaller than a decreasing function asymptotically approaching the value alpha = 1/2. We show that multiple giant components appear simultaneously in a strongly discontinuous percolation transition and remain distinct. Furthermore, tuning the value of alpha determines the number of such components with smaller alpha leading to an increasingly delayed and more explosive transition. The location of the critical point and strongly discontinuous nature are not affected if only edges which span components are sampled.