Rapid social connectivity

Itai Benjamini; Jonathan Hermon

doi:https://doi.org/10.1214/19-EJP294

Rapid social connectivity

Itai Benjamini, Jonathan Hermon

Department of Mathematics

Weizmann Institute of Science

Research output: Contribution to journal › Article › peer-review

Abstract

Given a graph G = (V, E), consider Poisson(vertical bar V vertical bar) 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 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 vertical bar V vertical bar

When G is regular the lower bound is improved to SC(G) >= log vertical bar V vertical bar - 6 log log vertical bar V vertical bar, with high probability. We determine SC(G) up to a constant factor in the cases that G is an expander and when it is the n-cycle.

Original language	English
Article number	32
Number of pages	33
Journal	Electronic Journal of Probability
Volume	24
DOIs	https://doi.org/10.1214/19-EJP294
State	Published - 2019

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title = "Rapid social connectivity",

abstract = "Given a graph G = (V, E), consider Poisson(vertical bar V vertical bar) 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 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 probabilityc log vertical bar V vertical barWhen G is regular the lower bound is improved to SC(G) >= log vertical bar V vertical bar - 6 log log vertical bar V vertical bar, with high probability. We determine SC(G) up to a constant factor in the cases that G is an expander and when it is the n-cycle.",

author = "Itai Benjamini and Jonathan Hermon",

note = "Acknowledgments. We thank Gady Kozma for his great contribution for some of the results presented in this paper. We like to thank also Riddhipratim Basu, Hilary Finucane, Eviatar Procaccia, Allan Sly and especially Ofer Zeitouni for many useful discussions and for reading some version of this work carefully and providing helpful comments.",

year = "2019",

doi = "https://doi.org/10.1214/19-EJP294",

language = "الإنجليزيّة",

volume = "24",

journal = "Electronic Journal of Probability",

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T1 - Rapid social connectivity

AU - Benjamini, Itai

AU - Hermon, Jonathan

N1 - Acknowledgments. We thank Gady Kozma for his great contribution for some of the results presented in this paper. We like to thank also Riddhipratim Basu, Hilary Finucane, Eviatar Procaccia, Allan Sly and especially Ofer Zeitouni for many useful discussions and for reading some version of this work carefully and providing helpful comments.

PY - 2019

Y1 - 2019

N2 - Given a graph G = (V, E), consider Poisson(vertical bar V vertical bar) 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 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 probabilityc log vertical bar V vertical barWhen G is regular the lower bound is improved to SC(G) >= log vertical bar V vertical bar - 6 log log vertical bar V vertical bar, with high probability. We determine SC(G) up to a constant factor in the cases that G is an expander and when it is the n-cycle.

AB - Given a graph G = (V, E), consider Poisson(vertical bar V vertical bar) 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 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 probabilityc log vertical bar V vertical barWhen G is regular the lower bound is improved to SC(G) >= log vertical bar V vertical bar - 6 log log vertical bar V vertical bar, with high probability. We determine SC(G) up to a constant factor in the cases that G is an expander and when it is the n-cycle.

U2 - https://doi.org/10.1214/19-EJP294

DO - https://doi.org/10.1214/19-EJP294

M3 - مقالة

SN - 1083-6489

VL - 24

JO - Electronic Journal of Probability

JF - Electronic Journal of Probability

M1 - 32

ER -

Rapid social connectivity

Abstract

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