Modulus consensus in discrete-time signed networks and properties of special recurrent inequalities

Recently the dynamics of signed networks, where the ties among the agents can be both positive (attractive) or negative (repulsive) have attracted substantial attention of the research community. Examples of such networks are models of opinion dynamics over signed graphs. It has been shown that under mild connectivity assumptions these protocols provide the convergence of opinions in absolute value, whereas their signs may differ. This “modulus consensus” may correspond to the bipartite consensus (the opinions split into two clusters, converging to two opposite values) or the asymptotic stability of the system (the opinions always converge to zero). In this paper, we demonstrate that the phenomenon of modulus consensus in a signed network is a manifestation of a more general, regarding the solutions of special recurrent inequalities, associated to conventional first-order consensus algorithms. Although such a recurrent inequality does not provide the uniqueness of a solution, it can be shown that, under some natural assumptions, each of its bounded solutions has a limit and, moreover, converges to consensus. A similar property has previously been established for special continuous-time differential inequalities in [1]. Besides analysis of signed networks, we link the consensus properties of recurrent inequalities to the convergence properties of distributed optimization algorithms and stability properties of substochastic matrices.