Consensus Problems on Networks With Antagonistic Interactions

TLDR

Consensus protocols achieve agreement through collaborative efforts represented by a communication graph with nonnegative weights, and the conditions for such agreement resemble those in monotone systems theory. This paper investigates whether consensus can be achieved when agents interact antagonistically, represented by negative weights on the communication graph. Linear and nonlinear Laplacian feedback designs are proposed. We show that on signed networks agents can converge to a consensus value up to a sign, and we provide necessary and sufficient conditions for this to occur.

Abstract

In a consensus protocol an agreement among agents is achieved thanks to the collaborative efforts of all agents, expresses by a communication graph with nonnegative weights. The question we ask in this paper is the following: is it possible to achieve a form of agreement also in presence of antagonistic interactions, modeled as negative weights on the communication graph? The answer to this question is affirmative: on signed networks all agents can converge to a consensus value which is the same for all agents except for the sign. Necessary and sufficient conditions are obtained to describe cases in which this is possible. These conditions have strong analogies with the theory of monotone systems. Linear and nonlinear Laplacian feedback designs are proposed.