Ordinary Percolation with Discontinuous Transitions

TL;DR

This paper demonstrates a novel type of percolation transition with an abrupt jump in the order parameter on small-world networks combining 1D lattices with long-range bonds, revealing features of explosive cluster growth.

Contribution

It provides a rigorous example of a discontinuous percolation transition on a small-world network with hierarchical long-range bonds.

Findings

Percolation transition occurs abruptly at a nontrivial critical point.

Order parameter jumps instantly to a finite value.

Features of explosive cluster growth are revealed.

Abstract

Percolation on a one-dimensional lattice and fractals such as the Sierpinski gasket is typically considered to be trivial because they percolate only at full bond density. By dressing up such lattices with small-world bonds, a novel percolation transition with explosive cluster growth can emerge at a nontrivial critical point. There, the usual order parameter, describing the probability of any node to be part of the largest cluster, jumps instantly to a finite value. Here, we provide a simple example of this transition in form of a small-world network consisting of a one-dimensional lattice combined with a hierarchy of long-range bonds that reveals many features of the transition in a mathematically rigorous manner.