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- W4298275678 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 $t_{lambda}(G)$." @default.
- W4298275678 created "2022-10-01" @default.
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- W4298275678 date "2016-10-13" @default.
- W4298275678 modified "2023-10-14" @default.
- W4298275678 title "The social network model on infinite graphs" @default.
- W4298275678 doi "https://doi.org/10.48550/arxiv.1610.04293" @default.
- W4298275678 hasPublicationYear "2016" @default.
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