# Mathematics Colloquia and Seminars

We consider the following model of a social network, in which people move on an infinite regular 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 discrete-time 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 with probability 1?