The social network model on infinite graphs

TL;DR

This paper studies a model of random walkers on infinite graphs, showing that on vertex-transitive amenable graphs all walkers eventually connect, while on non-amenable graphs a phase transition occurs with complex connectivity properties.

Contribution

It establishes the conditions under which walkers on infinite graphs become connected, identifying phase transitions and providing bounds on critical parameters.

Findings

On vertex-transitive amenable graphs, all walkers eventually connect.

On non-amenable graphs, a phase transition at some positive lambda exists.

In non-amenable graphs, finite time connectivity among infinite walkers occurs for all lambda.

Abstract

Given an infinite connected regular graph $G=(V,E)$, place at each vertex Pois($\lambda$) walkers performing independent lazy simple random walks on $G$ simultaneously. When two walkers visit the same vertex at the same time they are declared to be acquainted. We show that when $G$ is vertex-transitive and amenable, for all $\lambda>0$ a.s. any pair of walkers will eventually have a path of acquaintances between them. In contrast, we show that when $G$ is non-amenable (not necessarily transitive) there is always a phase transition at some $\lambda_{c}(G)>0$. We give general bounds on $\lambda_{c}(G)$ and study the case that $G$ is the $d$-regular tree in more details. Finally, we show that in the non-amenable setup, for every $\lambda$ there exists a finite time $t_{\lambda}(G)$ such that a.s. there exists an infinite set of walkers having a path of acquaintances between them by time $t_{\lambda}(G)$.