Component structure of the vacant set induced by a random walk on a random graph

TL;DR

This paper investigates the structure of the unvisited vertices (vacant set) in random walks on various random graphs, revealing phase transitions in component sizes and connectivity properties.

Contribution

It characterizes the phase transition of the vacant set's component structure on random graphs and provides explicit thresholds and properties for different graph models.

Findings

Existence of a phase transition in the vacant set's component structure.

Determination of the critical time t* for different graph models.

Description of the size and degree distribution of components at different times.

Abstract

We consider random walks on several classes of graphs and explore the likely structure of the vacant set, i.e. the set of unvisited vertices. Let \Gamma(t) be the subgraph induced by the vacant set of the walk at step t. We show that for random graphs G_{n,p} (above the connectivity threshold) and for random regular graphs G_r, r \geq 3, the graph \Gamma(t) undergoes a phase transition in the sense of the well-known Erdos-Renyi phase transition. Thus for t \leq (1-\epsilon)t^*, there is a unique giant component, plus components of size O(log n), and for t \geq (1+\epsilon)t^* all components are of size O(log n). For G_{n,p} and G_r we give the value of t^*, and the size of \Gamma(t). For G_r, we also give the degree sequence of \Gamma(t), the size of the giant component (if any) of \Gamma(t) and the number of tree components of \Gamma(t) of a given size k=O(log n). We also show that for random digraphs D_{n,p} above the strong connectivity threshold, there is a similar directed phase transition. Thus for t\leq (1-\epsilon)t^*, there is a unique strongly connected giant component, plus strongly connected components of size O(log n), and for t\geq (1+\epsilon)t^* all strongly connected components are of size O(log n).