Reaching a Consensus on Random Networks: The Power of Few

Authors Linh Tran, Van Vu

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Author Details

Linh Tran
  • Department of Mathematics, Yale University, New Haven, CT, USA
Van Vu
  • Department of Mathematics, Yale University, New Haven, CT, USA


We would like to thank A. Do, A, Ferber, A. Deneanu, J. Fox and X. Chen for inspiring discussions, H. V. Le, T. Can and L. T. D. Tran for proof-reading.

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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)


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.

Subject Classification

ACM Subject Classification
  • Mathematics of computing → Random graphs
  • Mathematics of computing → Graph theory
  • Mathematics of computing → Probability and statistics
  • Random Graphs Majority Dynamics Consensus


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