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

TL;DR

This paper investigates the emergent behaviors in signed random dynamical networks with a state-flipping model, revealing conditions for consensus and divergence in complex, time-varying social, biological, and engineering systems.

Contribution

It introduces a novel state-flipping model for signed networks with random interactions, analyzing convergence and divergence conditions under dynamic environments.

Findings

All links contribute to consensus of absolute node states.

Conditions for almost sure convergence and divergence are established.

Divergence occurs if the maximum network state diverges almost surely.

Abstract

Recent studies from social, biological, and engineering network systems have drawn attention to the dynamics over signed networks, where each link is associated with a positive/negative sign indicating trustful/mistrustful, activator/inhibitor, or secure/malicious interactions. We study asymptotic dynamical patterns that emerge among a set of nodes that interact in a dynamically evolving signed random network. Node interactions take place at random on a sequence of deterministic signed graphs. Each node receives positive or negative recommendations from its neighbors depending on the sign of the interaction arcs, and updates its state accordingly. Recommendations along a positive arc follow the standard consensus update. As in the work by Altafini, negative recommendations use an update where the sign of the neighbor state is flipped. Nodes may weight positive and negative recommendations differently, and random processes are introduced to model the time-varying attention that nodes pay to these recommendations. Conditions for almost sure convergence and divergence of the node states are established. We show that under this so-called state-flipping model, all links contribute to a consensus of the absolute values of the nodes, even under switching sign patterns and dynamically changing environment. A no-survivor property is established, indicating that every node state diverges almost surely if the maximum network state diverges.