Open Access
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 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 nonamenable (not necessarily transitive) there is always a phase transition at some λc(G)>0. We give general bounds on λc(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 λ there exists a finite time tλ(G) such that a.s. there exists an infinite set of walkers having a path of acquaintances between them by time tλ(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

Keywords: amenability , infinite cluster , percolation , phase transition , Random walks , Social network

Rights: Copyright © 2020 Institute of Mathematical Statistics

Vol.30 • No. 2 • April 2020
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