Ordering in voter models on networks: exact reduction to a single-coordinate diffusion

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Abstract

We study the voter model and related random copying processes on arbitrarily complex network structures. Through a representation of the dynamics as a particle reaction process, we show that a quantity measuring the degree of order in a finite system is, under certain conditions, exactly governed by a universal diffusion equation. Whenever this reduction occurs, the details of the network structure and random copying process affect only a single parameter in the diffusion equation. The validity of the reduction can be established with considerably less information than one might expect: it suffices to know just two characteristic timescales within the dynamics of a single pair of reacting particles. We develop methods to identify these timescales and apply them to deterministic and random network structures. We focus in particular on how the ordering time is affected by degree correlations, since such effects are difficult to access by existing theoretical approaches.

Original languageEnglish
Article number385003
Pages (from-to)-
Number of pages33
JournalJournal of physics a-Mathematical and theoretical
Volume43
Issue number38
DOIs
Publication statusPublished - 24 Sep 2010

Keywords

  • SMALL-WORLD NETWORKS
  • SUBDIVIDED POPULATION
  • COMPLEX NETWORKS
  • EFFECTIVE SIZE
  • MARKOV-CHAINS
  • ISLAND-MODEL
  • COALESCENT
  • MIGRATION
  • DYNAMICS
  • CONVERGENCE

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