Electronic Journal of Probability

Rapid social connectivity

Itai Benjamini and Jonathan Hermon

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Given a graph $G=(V,E)$, consider Poisson($|V|$) walkers performing independent lazy simple random walks on $G$ simultaneously, where the initial position of each walker is chosen independently with probability proportional to the degrees. When two walkers visit the same vertex at the same time they are declared to be acquainted. The social connectivity time $\mathrm{SC} (G)$ is defined as the first time in which there is a path of acquaintances between every pair of walkers. It is shown that when the average degree of $G$ is $d$, with high probability \[ c\log |V| \le \mathrm{SC} (G) \le C d^{1+5 \cdot 1_{G \text{ is not regular} } } \log ^3 |V|. \] When $G$ is regular the lower bound is improved to $\mathrm{SC} (G) \ge \log |V| -6 \log \log |V| $, with high probability. We determine $\mathrm{SC} (G)$ up to a constant factor in the cases that $G$ is an expander and when it is the $n$-cycle.

Article information

Electron. J. Probab., Volume 24 (2019), paper no. 32, 33 pp.

Received: 28 May 2018
Accepted: 17 March 2019
First available in Project Euclid: 9 April 2019

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 82C41: Dynamics of random walks, random surfaces, lattice animals, etc. [See also 60G50] 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43] 82B43: Percolation [See also 60K35] 60J10: Markov chains (discrete-time Markov processes on discrete state spaces) 05C81: Random walks on graphs 91D30: Social networks

social network random walks giant component coalescence process

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Benjamini, Itai; Hermon, Jonathan. Rapid social connectivity. Electron. J. Probab. 24 (2019), paper no. 32, 33 pp. doi:10.1214/19-EJP294. https://projecteuclid.org/euclid.ejp/1554775411

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