### Chuan Qin (University of California, Davis)

Title: Uniqueness of Friend Cluster in a Social Network on Non-Amenable Graphs

Abstract: We consider the following model of a social network, where people travel on an infinite graph $G$ and make friends. For each vertex $x$ in $G$, there are initially $N(x)$ people at $x$, where $N(x)$'s are i.i.d. Poisson random variables with mean $\lambda$. Each person performs a lazy simple random walk, independently of others. Whenever two people meet at a vertex, they befriend each other and each other's friends. We answer the following question asked by Itai Benjamini: For what values of $\lambda$ is it true that every pair of people eventually become friends almost surely? This is joint work with J. Hermon, B. Morris and A. Sly.