Conservation laws for the voter model in complex networks

TL;DR

This paper investigates the voter model on complex networks, revealing how different update rules affect magnetization conservation and the resulting metastable states, with implications for understanding consensus dynamics.

Contribution

It identifies how update rules influence magnetization conservation and metastability in the voter model on complex networks, highlighting a universal scaling law.

Findings

Node-update dynamics do not conserve average magnetization.

Link-update dynamics conserve average magnetization.

Scaling of metastable state lifetime is linear with system size under conservation.

Abstract

We consider the voter model dynamics in random networks with an arbitrary distribution of the degree of the nodes. We find that for the usual node-update dynamics the average magnetization is not conserved, while an average magnetization weighted by the degree of the node is conserved. However, for a link-update dynamics the average magnetization is still conserved. For the particular case of a Barabasi-Albert scale-free network the voter model dynamics leads to a partially ordered metastable state with a finite size survival time. This characteristic time scales linearly with system size only when the updating rule respects the conservation law of the average magnetization. This scaling identifies a universal or generic property of the voter model dynamics associated with the conservation law of the magnetization.