Given enough choice, simple local rules percolate discontinuously

TL;DR

This paper investigates simple local rules in percolation processes, demonstrating that increasing the number of candidate edges can induce discontinuous transitions in large networks, contrasting with previous results on continuous transitions.

Contribution

It introduces a minimal process based on candidate edge degrees that can produce discontinuous percolation transitions in the thermodynamic limit, expanding understanding of percolation phenomena.

Findings

Discontinuous transitions occur when candidate edges grow with network size.

Critical window size is proportional to n/k(n), shrinking with increasing k(n).

Bounded size rules always lead to continuous transitions, even with infinite choices.

Abstract

There is still much to discover about the mechanisms and nature of discontinuous percolation transitions. Much of the past work considers graph evolution algorithms known as Achlioptas processes in which a single edge is added to the graph from a set of $k$ randomly chosen candidate edges at each timestep until a giant component emerges. Several Achlioptas processes seem to yield a discontinuous percolation transition, but it was proven by Riordan and Warnke that the transition must be continuous in the thermodynamic limit. However, they also proved that if the number $k(n)$ of candidate edges increases with the number of nodes, then the percolation transition may be discontinuous. Here we attempt to find the simplest such process which yields a discontinuous transition in the thermodynamic limit. We introduce a process which considers only the degree of candidate edges and not component size. We calculate the critical point $t_{c}=(1-\theta(\frac{1}{k}))n$ and rigorously show that the critical window is of size $O(\frac{n}{k(n)})$. If $k(n)$ grows very slowly, for example $k(n)=\log n$, the critical window is barely sublinear and hence the phase transition is discontinuous but appears continuous in finite systems. We also present arguments that Achlioptas processes with bounded size rules will always have continuous percolation transitions even with infinite choice.