Voter model on heterogeneous directed networks

TL;DR

This paper analyzes the consensus time of the voter model on large, heterogeneous, directed networks with power-law degree distributions, deriving explicit formulas and validating them through simulations.

Contribution

It extends mean-field predictions to directed heavy-tailed networks, providing the first explicit consensus-time formula in this setting.

Findings

Derived exact asymptotics for consensus time in directed heavy-tailed networks.

Validated predictions through extensive simulations across various heterogeneity regimes.

Identified conditions under which Wright-Fisher diffusive behavior emerges or breaks down.

Abstract

We investigate the consensus dynamics of the voter model on large random graphs with heterogeneous and directed features, focusing in particular on networks with power-law degree distributions. By extending recent results on sparse directed graphs, we derive exact first-order asymptotics for the expected consensus time in directed configuration models with i.i.d. Pareto-distributed in- and out-degrees. For any tail exponent {\alpha}>0, we derive the mean consensus time scaling depending on the network size and a pre-factor that encodes detailed structural properties of the degree sequences. We give an explicit description of the pre factor in the directed setting. This extends and sharpens previous mean-field predictions from statistical physics, providing the first explicit consensus-time formula in the directed heavy-tailed setting. Through extensive simulations, we confirm the validity of our predictions across a wide range of heterogeneity regimes, including networks with infinite variance and infinite mean degree distribution. We further explore the interplay between network topology and voter dynamics, highlighting how degree fluctuations and maximal degrees shape the consensus landscape. Complementing the asymptotic analysis, we provide numerical evidence for the emergence of Wright-Fisher diffusive behavior in both directed and undirected ensembles under suitable mixing conditions, and demonstrate the breakdown of this approximation in the in the infinite mean regime.