Conservation laws for voter-like models on directed networks

TL;DR

This paper analytically investigates conservation laws in voter-like models on directed networks, revealing how certain weighted combinations of states remain constant and influence the system's evolution towards absorbing states.

Contribution

It derives explicit conservation laws for voter and invasion processes on directed networks, including for correlated and uncorrelated cases, using a heterogeneous mean field approach.

Findings

Conservation laws exist for all three processes studied.

The conserved quantities depend on local degree properties.

Time evolution decays exponentially to a stationary state.

Abstract

We study the voter model, under node and link update, and the related invasion process on a single strongly connected component of a directed network. We implement an analytical treatment in the thermodynamic limit using the heterogeneous mean field assumption. From the dynamical rules at the microscopic level, we find the equations for the evolution of the relative densities of nodes in a given state on heterogeneous networks with arbitrary degree distribution and degree-degree correlations. We prove that conserved quantities as weighted linear superpositions of spin states exist for all three processes and, for uncorrelated directed networks, we derive their specific expressions. We also discuss the time evolution of the relative densities that decay exponentially to a homogeneous stationary value given by the conserved quantity. The conservation laws obtained in the thermodynamic limit for a system that does not order in that limit determine the probabilities of reaching the absorbing state for a finite system. The contribution of each degree class to the conserved quantity is determined by a local property. Depending on the dynamics, the highest contribution is associated to influential nodes reaching a large number of outgoing neighbors, not too influenceable ones with a low number of incoming connections, or both at the same time.