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Asynchronous Majority Dynamics on Binomial Random Graphs

Authors Divyarthi Mohan , Paweł Prałat

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Author Details

Divyarthi Mohan
  • Blavatnik School of Computer Science, Tel Aviv University, Israel
Paweł Prałat
  • Department of Mathematics, Toronto Metropolitan University, Canada

Funding

  • Mohan, Divyarthi: This project was funded in part by the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation program (grant agreement no. 866132) and by the Israel Science Foundation (grant no. 317/17).

Acknowledgements

This work was done while the authors were visiting the Simons Institute for the Theory of Computing.

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Divyarthi Mohan and Paweł Prałat. Asynchronous Majority Dynamics on Binomial Random Graphs. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 317, pp. 5:1-5:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)
https://doi.org/10.4230/LIPIcs.APPROX/RANDOM.2024.5

Abstract

We study information aggregation in networks when agents interact to learn a binary state of the world. Initially each agent privately observes an independent signal which is correct with probability 1/2+δ for some δ > 0. At each round, a node is selected uniformly at random to update their public opinion to match the majority of their neighbours (breaking ties in favour of their initial private signal). Our main result shows that for sparse and connected binomial random graphs G(n,p) the process stabilizes in a correct consensus in 𝒪(nlog² n/log log n) steps with high probability. In fact, when log n/n ≪ p = o(1) the process terminates at time T^ = (1+o(1))nlog n, where T^ is the first time when all nodes have been selected at least once. However, in dense binomial random graphs with p = Ω(1), there is an information cascade where the process terminates in the incorrect consensus with probability bounded away from zero.

Subject Classification

ACM Subject Classification
  • Mathematics of computing → Stochastic processes
  • Mathematics of computing → Discrete mathematics
  • Theory of computation → Social networks
Keywords
  • Opinion dynamics
  • Social learning
  • Stochastic processes
  • Random Graphs
  • Consensus

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