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

Recent studies from social, biological, and engineering network systems have drawn at-tention to the dynamics over signed networks, where each link is associated with a pos-itive/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 neg-ative 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 con-sensus 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 recommen-dations differently, and random processes are introduced to model the time-varying attention that nodes pay to these recommendations. Conditions for almost sure convergence and di-vergence 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