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Central limit theorem for majority dynamics: Bribing three voters suffices

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  • Berkowitz, Ross
  • Devlin, Pat

Abstract

Given a graph G and some initial labeling σ:V(G)→{Red,Blue} of its vertices, the majority dynamics model is the deterministic process where at each stage, every vertex simultaneously replaces its label with the majority label among its neighbors (remaining unchanged in the case of a tie). We prove—for a wide range of parameters—that if an initial assignment is fixed and we independently sample an Erdős–Rényi random graph, Gn,p, then after one step of majority dynamics, the number of vertices of each label follows a central limit law. As a corollary, we provide a strengthening of a theorem of Benjamini, Chan, O’Donnell, Tamuz, and Tan about the number of steps required for the process to reach unanimity when the initial assignment is also chosen randomly.

Suggested Citation

  • Berkowitz, Ross & Devlin, Pat, 2022. "Central limit theorem for majority dynamics: Bribing three voters suffices," Stochastic Processes and their Applications, Elsevier, vol. 146(C), pages 187-206.
    • Handle: RePEc:eee:spapps:v:146:y:2022:i:c:p:187-206
    DOI: 10.1016/j.spa.2022.01.010
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    References listed on IDEAS

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    1. Venkatesh Bala & Sanjeev Goyal, 1998. "Learning from Neighbours," The Review of Economic Studies, Review of Economic Studies Ltd, vol. 65(3), pages 595-621.
    2. Benjamini, Itai & Chan, Siu-On & O’Donnell, Ryan & Tamuz, Omer & Tan, Li-Yang, 2016. "Convergence, unanimity and disagreement in majority dynamics on unimodular graphs and random graphs," Stochastic Processes and their Applications, Elsevier, vol. 126(9), pages 2719-2733.
    3. Ellison, Glenn & Fudenberg, Drew, 1993. "Rules of Thumb for Social Learning," Journal of Political Economy, University of Chicago Press, vol. 101(4), pages 612-643, August.
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