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April 2020 The social network model on infinite graphs
Jonathan Hermon, Ben Morris, Chuan Qin, Allan Sly
Ann. Appl. Probab. 30(2): 902-935 (April 2020). DOI: 10.1214/19-AAP1520

Abstract

Given an infinite connected regular graph $G=(V,E)$, place at each vertex $\operatorname{Poisson}(\lambda)$ 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 nonamenable (not necessarily transitive) there is always a phase transition at some $\lambda_{\mathrm{c}}(G)>0$. We give general bounds on $\lambda_{\mathrm{c}}(G)$ and study the case that $G$ is the $d$-regular tree in more detail. Finally, we show that in the nonamenable 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 $t_{\lambda}(G)$.

Citation

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Jonathan Hermon. Ben Morris. Chuan Qin. Allan Sly. "The social network model on infinite graphs." Ann. Appl. Probab. 30 (2) 902 - 935, April 2020. https://doi.org/10.1214/19-AAP1520

Information

Received: 1 September 2018; Revised: 1 June 2019; Published: April 2020
First available in Project Euclid: 8 June 2020

zbMATH: 07236138
MathSciNet: MR4108126
Digital Object Identifier: 10.1214/19-AAP1520

Subjects:
Primary: 60G50, 60J10, 60K35
Secondary: 82B43, 82C41

Rights: Copyright © 2020 Institute of Mathematical Statistics

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Vol.30 • No. 2 • April 2020
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