Given an infinite connected regular graph G = (V, E), place at each vertex Poisson(λ) 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 λ > 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 λ c (G) > 0. We give general bounds on λ c (G) and study the case that G is the d-regular tree in more detail. Finally, we show that in the non-amenable setup, for every λ there exists a finite time t λ (G) such that a.s. there exists an infinite set of walkers having a path of acquaintances between them by time t λ (G).
It is shown in REF that when G is vertex-transitive and amenable, for all λ > 0 a.s.
The social network model on infinite graphs
arXiv: Probability
Mathematics