## Electronic Journal of Probability

### Rapid social connectivity

#### Abstract

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

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

Dates
Accepted: 17 March 2019
First available in Project Euclid: 9 April 2019

https://projecteuclid.org/euclid.ejp/1554775411

Digital Object Identifier
doi:10.1214/19-EJP294

Mathematical Reviews number (MathSciNet)
MR3940762

Zentralblatt MATH identifier
07055670

#### Citation

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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