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We study asymptotic dynamical patterns that emerge among a set of nodes thatinteract in a dynamically evolving signed random network. Node interactionstake place at random on a sequence of deterministic signed graphs. Each nodereceives positive or negative recommendations from its neighbors depending onthe sign of the interaction arcs, and updates its state accordingly. Positiverecommendations follow the standard consensus update while two types ofnegative recommendations, each modeling a different type of antagonistic ormalicious interaction, are considered. Nodes may weigh positive and negativerecommendations differently, and random processes are introduced to model thetime-varying attention that nodes pay to the positive and negativerecommendations. Various conditions for almost sure convergence, divergence,and clustering of the node states are established. Some fundamentalsimilarities and differences are established for the two notions of negativerecommendations.

Emergent Behaviors over Signed Random Networks in Dynamical Environments