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DOI: 10.4230/LIPIcs.APPROX/RANDOM.2020.20

Reaching a Consensus on Random Networks: The Power of Few

Authors: Linh Tran and Van Vu

Published in: LIPIcs, Volume 176, Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2020)

Abstract

A community of n individuals splits into two camps, Red and Blue. The individuals are connected by a social network, which influences their colors. Everyday, each person changes his/her color according to the majority of his/her neighbors. Red (Blue) wins if everyone in the community becomes Red (Blue) at some point. We study this process when the underlying network is the random Erdos-Renyi graph G(n, p). With a balanced initial state (n/2 persons in each camp), it is clear that each color wins with the same probability. Our study reveals that for any constants p and ε, there is a constant c such that if one camp has n/2 + c individuals at the initial state, then it wins with probability at least 1 - ε. The surprising fact here is that c does not depend on n, the population of the community. When p = 1/2 and ε = .1, one can set c = 6, meaning one camp has n/2 + 6 members initially. In other words, it takes only 6 extra people to win an election with overwhelming odds. We also generalize the result to p = p_n = o(1) in a separate paper.

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Linh Tran and Van Vu. Reaching a Consensus on Random Networks: The Power of Few. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 176, pp. 20:1-20:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)

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@InProceedings{tran_et_al:LIPIcs.APPROX/RANDOM.2020.20,
  author =	{Tran, Linh and Vu, Van},
  title =	{{Reaching a Consensus on Random Networks: The Power of Few}},
  booktitle =	{Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2020)},
  pages =	{20:1--20:15},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-164-1},
  ISSN =	{1868-8969},
  year =	{2020},
  volume =	{176},
  editor =	{Byrka, Jaros{\l}aw and Meka, Raghu},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX/RANDOM.2020.20},
  URN =		{urn:nbn:de:0030-drops-126239},
  doi =		{10.4230/LIPIcs.APPROX/RANDOM.2020.20},
  annote =	{Keywords: Random Graphs Majority Dynamics Consensus}
}

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Reaching a Consensus on Random Networks: The Power of Few

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