Predictable Dynamics of Opinion Forming for Networks With Antagonistic Interactions

Abstract

Abstract-For communities of agents which are not necessarily cooperating, distributed processes of opinion forming are naturally represented by signed graphs, with positive edges representing friendly and cooperative interactions and negative edges the corresponding antagonistic counterpart. Unlike for nonnegative graphs, the outcome of a dynamical system evolving on a signed graph is not obvious and it is in general difficult to characterize, even when the dynamics are linear. In this paper we identify a significant class of signed graphs for which the linear dynamics are however predictable and show many analogies with positive dynamical systems. These cases correspond to adjacency matrices that are eventually positive, for which the PerronFrobenius property still holds and implies the existence of an invariant cone contained inside the positive orthant. As examples of applications, we determine cases in which it is possible to anticipate or impose unanimity of opinion in decision/voting processes even in presence of stubborn agents, and show how it is possible to extend the PageRank algorithm to include negative links.

Representing externally positive systems through minimal eventually positive realizations

“…The scope of this paper is to suggest a class of minimal realizations which are not positive but eventually positive in the sense that the state update matrix is allowed to have some negative entries but it becomes positive after a certain power [1], [8]. These matrices still satisfy the Perron-Frobenius theorem, with both left and right dominant eigenvector in R n + .…”

Spectral Conditions for Stability and Stabilization of Positive Equilibria for a Class of Nonlinear Cooperative Systems

“…From the results of [2], [3], [12], is clear that in this case the phase plane of the positive orthant is replicated in the negative orthant (R n − ). It remains however to understand to what extent the spectral conditions developed here can be extended beyond the positive/negative orthant.…”

Minimal eventually positive realizations of externally positive systems

“…Corollary 1 (Altafini & Lini (2015), Corollary 2) A ∈ PF n if and only if ∃ a proper polyhedral A-invariant cone K for which A is K -positive, and…”

“…If we relax the assumption of positivity of A while maintaining the condition that the eigenvector must be contained in R n + , then we still have that the free evolution of the state of a minimal realization becomes positive after a transient. Matrices A having both left and right dominant eigenvector in R n + form a special class of matrices called eventually positive, see Altafini & Lini (2015); Noutsos (2006). While these matrices can have negative entries (hence they do not correspond to positive realizations), they have the property that after a certain power they become positive matrices.…”