  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 $t_{\lambda}(G)$.
