Dynamics of link states in complex networks: The case of a majority rule

TL;DR

This paper investigates the dynamics of link states in complex networks using a majority rule, revealing diverse asymptotic configurations and mechanisms distinct from traditional node-based dynamics.

Contribution

It introduces a prototype model for link state dynamics, analyzing its behavior across different network topologies and uncovering novel asymptotic configurations.

Findings

Diverse nontrivial asymptotic configurations identified.

Mechanisms leading to configurations characterized.

Asymptotic states often differ from node dynamics counterparts.

Abstract

Motivated by the idea that some characteristics are specific to the relations between individuals and not of the individuals themselves, we study a prototype model for the dynamics of the states of the links in a fixed network of interacting units. Each link in the network can be in one of two equivalent states. A majority link-dynamics rule is implemented, so that in each dynamical step the state of a randomly chosen link is updated to the state of the majority of neighboring links. Nodes can be characterized by a link heterogeneity index, giving a measure of the likelihood of a node to have a link in one of the two states. We consider this link-dynamics model on fully connected networks, square lattices and Erd \"os-Renyi random networks. In each case we find and characterize a number of nontrivial asymptotic configurations, as well as some of the mechanisms leading to them and the time evolution of the link heterogeneity index distribution. For a fully connected network and random networks there is a broad distribution of possible asymptotic configurations. Most asymptotic configurations that result from link-dynamics have no counterpart under traditional node dynamics in the same topologies.