Consensus and voting on large graphs: An application of graph limit theory

Barton E. Lee

doi:10.3934/dcds.2018071

Article Contents

2018, Volume 38, Issue 4: 1719-1744. Doi: 10.3934/dcds.2018071

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Consensus and voting on large graphs: An application of graph limit theory

Barton E. Lee

School of Mathematics and Statistics, The University of New South Wales, Sydney NSW 2052, Australia

Received: May 2016

Revised: October 2017

Published: July 2018

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Abstract

Building on recent work by Medvedev (2014) we establish new connections between a basic consensus model, called the voting model, and the theory of graph limits. We show that in the voting model if consensus is attained in the continuum limit then solutions to the finite model will eventually be close to a constant function, and a class of graph limits which guarantee consensus is identified. It is also proven that the dynamics in the continuum limit can be decomposed as a direct sum of dynamics on the connected components, using Janson's definition of connectivity for graph limits. This implies that without loss of generality it may be assumed that the continuum voting model occurs on a connected graph limit.

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Mathematics Subject Classification: Primary: 45J05, 45M05; Secondary: 93C15, 93C20, 94C15.

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Figure 1. A plot of the function $W$

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