Emergent Behaviors over Signed Random Dynamical Networks: Relative-State-Flipping Model

TL;DR

This paper investigates the long-term behaviors of nodes in a dynamically changing signed network, where positive links promote consensus and negative links induce relative-state flipping, providing conditions for convergence, divergence, and clustering.

Contribution

It introduces a novel relative-state-flipping model with stochastic attentions and deterministic weights, extending previous models and analyzing emergent behaviors in signed random networks.

Findings

Conditions for almost sure convergence and divergence

A new criterion for almost sure state clustering

Differences highlighted between relative-state-flipping and state-flipping models

Abstract

We study asymptotic dynamical patterns that emerge among a set of nodes interacting in a dynamically evolving signed random network, where positive links carry out standard consensus and negative links induce relative-state flipping. A sequence of deterministic signed graphs define potential node interactions that take place independently. Each node receives a positive recommendation consistent with the standard consensus algorithm from its positive neighbors, and a negative recommendation defined by relative-state flipping from its negative neighbors. After receiving these recommendations, each node puts a deterministic weight to each recommendation, and then encodes these weighted recommendations in its state update through stochastic attentions defined by two Bernoulli random variables. We establish a number of conditions regarding almost sure convergence and divergence of the node states. We also propose a condition for almost sure state clustering for essentially weakly balanced graphs, with the help of several martingale convergence lemmas. Some fundamental differences on the impact of the deterministic weights and stochastic attentions to the node state evolution are highlighted between the current relative-state-flipping model and the state-flipping model considered in Altafini 2013 and Shi et al. 2014.