License: Creative Commons Attribution 3.0 Unported license (CC BY 3.0)
When quoting this document, please refer to the following
DOI: 10.4230/LIPIcs.APPROX/RANDOM.2020.20
URN: urn:nbn:de:0030-drops-126239
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Tran, Linh ; Vu, Van

Reaching a Consensus on Random Networks: The Power of Few

LIPIcs-APPROX20.pdf (0.5 MB)


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.

BibTeX - Entry

  author =	{Linh Tran and Van Vu},
  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 =	{Jaros{\l}aw Byrka and Raghu Meka},
  publisher =	{Schloss Dagstuhl--Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{},
  URN =		{urn:nbn:de:0030-drops-126239},
  doi =		{10.4230/LIPIcs.APPROX/RANDOM.2020.20},
  annote =	{Keywords: Random Graphs Majority Dynamics Consensus}

Keywords: Random Graphs Majority Dynamics Consensus
Collection: Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2020)
Issue Date: 2020
Date of publication: 11.08.2020
Supplementary Material: The simulation source code for the random process described in the paper is available at

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