A Hamiltonian approach for explosive percolation

TL;DR

This paper presents a Hamiltonian-based model linking equilibrium statistical mechanics to explosive percolation, identifying key factors for abrupt transitions and providing an exact solution in a specific limit.

Contribution

It introduces a novel cluster growth process that elucidates the conditions for first-order percolation transitions and offers an exact solution in a limiting case.

Findings

Abrupt (first-order) transition occurs when cluster sizes are kept similar.

Merging bonds dominate over redundant bonds for explosive percolation.

An exact solution is possible in the limit of only merging bonds using tree-like graphs.

Abstract

We introduce a cluster growth process that provides a clear connection between equilibrium statistical mechanics and an explosive percolation model similar to the one recently proposed by Achlioptas et al. [Science 323, 1453 (2009)]. We show that the following two ingredients are essential for obtaining an abrupt (first-order) transition in the fraction of the system occupied by the largest cluster: (i) the size of all growing clusters should be kept approximately the same, and (ii) the inclusion of merging bonds (i.e., bonds connecting vertices in different clusters) should dominate with respect to the redundant bonds (i.e., bonds connecting vertices in the same cluster). Moreover, in the extreme limit where only merging bonds are present, a complete enumeration scheme based on tree-like graphs can be used to obtain an exact solution of our model that displays a first-order transition. Finally, the proposed mechanism can be viewed as a generalization of standard percolation that discloses an entirely new family of models with potential application in growth and fragmentation processes of real network systems.