  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.
