  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, recently introduced by Altafini
(2012,2013) and extended to discrete-time case by Meng et al. (2014). 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 polarization of the opinions (or
bipartite consensus, including the usual consensus as a special case), or their
convergence to zero. In this paper, we demonstrate that the phenomenon of
modulus consensus in the discrete-time Altafini model is a manifestation of a
more general and profound fact, regarding the solutions of a special recurrent
inequality. 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 (Proskurnikov, Cao, 2016). Besides
analysis of signed networks, we link the consensus properties of recurrent
inequalities to the convergence analysis of distributed optimization algorithms
and the problems of Schur stability of substochastic matrices.
