Percolation through Isoperimetry

TL;DR

This paper establishes a tight isoperimetric condition on regular graphs that predicts the emergence of a giant component in percolated graphs, similar to Erdős–Rényi models, and extends the analysis to controlled expansion scenarios.

Contribution

It provides a precise isoperimetric criterion for phase transition in percolation on regular graphs, including tightness results and generalizations to partial expansion controls.

Findings

Giant component emerges around p=1/d under isoperimetric conditions

Condition is shown to be tight for the phase transition

Extension to partial expansion scenarios with size-controlled sets

Abstract

We provide a sufficient condition on the isoperimetric properties of a regular graph $G$ of growing degree $d$, under which the random subgraph $G_p$ typically undergoes a phase transition around $p=\frac{1}{d}$ which resembles the emergence of a giant component in the binomial random graph model $G(n,p)$. We further show that this condition is tight. More precisely, let $d=\omega(1)$, let $\epsilon>0$ be a small enough constant, and let $p \cdot d=1+\epsilon$. We show that if $C$ is sufficiently large and $G$ is a $d$-regular $n$-vertex graph where every subset $S\subseteq V(G)$ of order at most $\frac{n}{2}$ has edge-boundary of size at least $C|S|$, then $G_p$ typically has a unique linear sized component, whose order is asymptotically $y(\epsilon)n$, where $y(\epsilon)$ is the survival probability of a Galton-Watson tree with offspring distribution Po$(1+\epsilon)$. We further give examples to show that this result is tight both in terms of its dependence on $C$, and with respect to the order of the second-largest component. We also consider a more general setting, where we only control the expansion of sets up to size $k$. In this case, we show that if $G$ is such that every subset $S\subseteq V(G)$ of order at most $k$ has edge-boundary of size at least $d|S|$ and $p$ is such that $p\cdot d \geq 1 + \epsilon$, then $G_p$ typically contains a component of order $\Omega(k)$.