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

Abstract

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 . 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.

Differential inequalities in multi-agent coordination and opinion dynamics modeling

“…The dichotomy criteria allow to analyze stability of many protocols for target aggregation, containment control, target surrounding and distributed optimization in a unified way. The results of this paper can be extended to discrete-time, or recurrent inequalities x(t + 1) ≤ A(t)x(t), where A(t) are row-stochastic matrices (Proskurnikov and Cao, 2017). Their extensions to the delayed inequalities and inequalities of second and higher orders are subject of ongoing research.…”

Dynamic social balance and convergent appraisals via homophily and influence mechanisms

“…Since X ij (t + 1) = k W ik (t)X jk (t), each |W ik (t)| represents how much weight individual i assigns to the agreement on the appraisal of individual k. Note that the entry-wise absolute value, i.e., |W (t)|, is row-stochastic. Such type of influence matrices has been widely studied in the opinion dynamics with antagonism, see [18,37,31].…”

Opinion Dynamics With Competitive Relationship and Switching Topologies

“…In Theorem 4, we assume that cooperation and competition exist at the same time. On the other hand, in [17], [55], DeGroot model with dynamic changing topologies and competition was examined where the structurally balanced and unbalanced network topologies were considered. It should be pointed out the diagonal entries of the adjacency matrix are not allowed to take negative values in [17], [55].…”