Voter models on weighted networks

TL;DR

This paper investigates how voter and Moran processes behave on weighted complex networks, deriving analytical models and phase diagrams that reveal how edge weights influence consensus times and dynamics.

Contribution

It introduces a heterogeneous mean-field approach for weighted networks and analyzes the impact of degree-dependent weights on voter model dynamics.

Findings

Derived phase diagrams showing different scaling laws for consensus time.

Heterogeneous mean-field approach accurately predicts dynamics for small |theta|.

Numerical simulations confirm analytical results for certain parameter ranges.

Abstract

We study the dynamics of the voter and Moran processes running on top of complex network substrates where each edge has a weight depending on the degree of the nodes it connects. For each elementary dynamical step the first node is chosen at random and the second is selected with probability proportional to the weight of the connecting edge. We present a heterogeneous mean-field approach allowing to identify conservation laws and to calculate exit probabilities along with consensus times. In the specific case when the weight is given by the product of nodes' degree raised to a power theta, we derive a rich phase-diagram, with the consensus time exhibiting various scaling laws depending on theta and on the exponent of the degree distribution gamma. Numerical simulations give very good agreement for small values of |theta|. An additional analytical treatment (heterogeneous pair approximation) improves the agreement with numerics, but the theoretical understanding of the behavior in the limit of large |theta| remains an open challenge.