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Voter model on heterogeneous graphs.

V Sood1, S Redner

  • 1Theory Division and Center for Nonlinear Studies, Los Alamos National Laboratory, Los Alamos, New Mexico 87545, USA. vsood@bu.edu

Physical Review Letters
|May 21, 2005
PubMed
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The voter model on heterogeneous graphs shows consensus time depends on network structure. Specifically, the mean consensus time T(N) scales with network size N and degree distribution moments, offering insights into opinion dynamics.

Area of Science:

  • Statistical physics
  • Network science
  • Computational social science

Background:

  • The voter model is a fundamental tool for studying opinion dynamics and consensus formation in social networks.
  • Heterogeneous graphs, characterized by diverse node connectivity, are more realistic representations of many real-world systems.

Purpose of the Study:

  • To analyze the impact of graph heterogeneity on the time it takes to reach consensus in the voter model.
  • To derive analytical expressions for the mean consensus time based on network properties.

Main Methods:

  • Utilizing the nonconservation of magnetization as a key observable to track consensus.
  • Deriving scaling relations for the mean consensus time T(N) as a function of network size N and degree distribution moments.
  • Investigating specific cases, including power-law degree distributions.

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Main Results:

  • The mean consensus time T(N) scales as N*mu(2)^(1/mu(2)) for arbitrary uncorrelated degree distributions, where mu(k) is the k-th moment.
  • For power-law distributions n(k) ~ k^(-nu), distinct scaling behaviors emerge: N for nu > 3, N/ln(N) for nu = 3, N^((2nu-4)/(nu-1)) for 2 < nu < 3, (lnN)^2 for nu = 2, and O(1) for nu < 2.
  • The derived scaling laws were validated against simulation data for both uncorrelated and correlated networks.

Conclusions:

  • Network degree distribution critically influences the speed of consensus formation in the voter model.
  • The study provides a detailed theoretical framework for understanding consensus dynamics on complex, heterogeneous networks.
  • These findings have implications for modeling information diffusion, opinion spreading, and collective behavior in diverse systems.