Electronic Journal of Probability

Rapid social connectivity

Itai Benjamini and Jonathan Hermon

Full-text: Open access

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

Permanent link to this document
https://projecteuclid.org/euclid.ejp/1554775411

Digital Object Identifier
doi:10.1214/19-EJP294

Mathematical Reviews number (MathSciNet)
MR3940762

Zentralblatt MATH identifier
07055670

Subjects
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

Keywords
social network random walks giant component coalescence process

Rights
Creative Commons Attribution 4.0 International License.

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


Export citation

References

  • [1] David Aldous and Jim Fill, Reversible Markov chains and random walks on graphs, 2002, Unfinished manuscript. Available at http://www.stat.berkeley.edu/~aldous/RWG/book.html.
  • [2] Noga Alon, Itai Benjamini, and Alan Stacey, Percolation on finite graphs and isoperimetric inequalities, Ann. Probab. 32 (2004), no. 3A, 1727–1745.
  • [3] O. S. M. Alves, F. P. Machado, and S. Yu. Popov, Phase transition for the frog model, Electron. J. Probab. 7 (2002), no. 16, 21.
  • [4] O. S. M. Alves, F. P. Machado, and S. Yu. Popov, The shape theorem for the frog model, Ann. Appl. Probab. 12 (2002), no. 2, 533–546.
  • [5] Itai Benjamini, Luiz Renato Fontes, Jonathan Hermon, and Fabio Pabio Machado, On an epidemic model on finite graphs, Arxiv preprint arXiv:1610.04301 (2016).
  • [6] Lucas Boczkowski, Yuval Peres, and Perla Sousi, Sensitivity of mixing times in Eulerian digraphs, SIAM J. Discrete Math. 32 (2018), no. 1, 624–655.
  • [7] Devdatt Dubhashi and Desh Ranjan, Balls and bins: a study in negative dependence, Random Structures Algorithms 13 (1998), no. 2, 99–124.
  • [8] Sharad Goel, Ravi Montenegro, and Prasad Tetali, Mixing time bounds via the spectral profile, Electron. J. Probab. 11 (2006), no. 1, 1–26.
  • [9] Jonathan Hermon, Frogs on trees?, Electron. J. Probab. 23 (2018), Paper No. 17, 40.
  • [10] Jonathan Hermon, Ben Morris, Chuan Qin, and Allan Sly, The social network model on infinite graphs, Arxiv preprint arXiv:1610.04293 (2016).
  • [11] Jonathan Hermon and Richard Pymar, The exclusion process mixes (almost) faster than independent particles, arXiv preprint arXiv:1808.10846 (2018).
  • [12] Christopher Hoffman, Tobias Johnson, and Matthew Junge, Recurrence and transience for the frog model on trees, Ann. Probab. 45 (2017), no. 5, 2826–2854.
  • [13] Olav Kallenberg, Foundations of modern probability, second ed., Probability and its Applications (New York), Springer-Verlag, New York, 2002.
  • [14] Harry Kesten and Vladas Sidoravicius, The spread of a rumor or infection in a moving population, Ann. Probab. 33 (2005), no. 6, 2402–2462.
  • [15] Harry Kesten and Vladas Sidoravicius, A shape theorem for the spread of an infection, Ann. of Math. 167 (2008), no. 3, 701–766.
  • [16] David A. Levin, Yuval Peres, and Elizabeth L. Wilmer, Markov chains and mixing times, American Mathematical Society, Providence, RI, 2017, Second edition of [MR2466937], With a chapter on “Coupling from the past” by James G. Propp and David B. Wilson.
  • [17] S. Yu. Popov, Frogs in random environment, J. Statist. Phys. 102 (2001), no. 1-2, 191–201.
  • [18] András Telcs and Nicholas C. Wormald, Branching and tree indexed random walks on fractals, J. Appl. Probab. 36 (1999), no. 4, 999–1011.