Consensus and Voting on Large Graphs: An Application of Graph Limit Theory

TL;DR

This paper links the consensus voting model on large graphs to graph limit theory, showing how continuum limits predict finite graph behavior and identifying conditions for consensus.

Contribution

It establishes new theoretical connections between the voting model and graph limits, including decomposition of dynamics and conditions for consensus in large networks.

Findings

Consensus in the continuum limit implies finite models approach a constant function.

Graph limits guaranteeing consensus are characterized.

Dynamics can be decomposed based on connectivity of graph limits.

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.