  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.
