Deriving an underlying mechanism for discontinuous percolation

TL;DR

This paper introduces a stochastic graph evolution model demonstrating that growth by overtaking causes discontinuous percolation transitions, with the transition nature depending on a parameter, and reveals unique component size distributions without finite size effects.

Contribution

The study derives a simple mechanism, growth by overtaking, responsible for discontinuous percolation, and characterizes how the transition type depends on a key parameter.

Findings

Discontinuous transition driven by overtaking for β<1

Weakly discontinuous transition for β>1

Unique component size distributions with no finite size effects

Abstract

Understanding what types of phenomena lead to discontinuous phase transitions in the connectivity of random networks is an outstanding challenge. Here we show that a simple stochastic model of graph evolution leads to a discontinuous percolation transition and we derive the underlying mechanism responsible: growth by overtaking. Starting from a collection of $n$ isolated nodes, potential edges chosen uniformly at random from the complete graph are examined one at a time while a cap, $k$, on the maximum allowed component size is enforced. Edges whose addition would exceed $k$ can be simply rejected provided the accepted fraction of edges never becomes smaller than a function which decreases with $k$ as $g(k) = 1/2 + (2k)^{-\beta}$. We show that if $\beta < 1$ it is always possible to reject a sampled edge and the growth in the largest component is dominated by an overtaking mechanism leading to a discontinuous transition. If $\beta > 1$, once $k \ge n^{1/\beta}$, there are situations when a sampled edge must be accepted leading to direct growth dominated by stochastic fluctuations and a "weakly" discontinuous transition. We also show that the distribution of component sizes and the evolution of component sizes are distinct from those previously observed and show no finite size effects for the range of $\beta$ studied.