Multiequilibria Analysis for a Class of Collective Decision-Making Networked Systems

The models of collective decision-making considered in this paper are nonlinear-interconnected cooperative systems with saturating interactions. These systems encode the possible outcomes of a decision process into different steady states of the dynamics. In particular, they are characterized by two main attractors in the positive and negative orthants, representing two choices of agreement among the agents, associated with the Perron-Frobenius eigenvector of the system. In this paper, we give conditions for the appearance of other equilibria of mixed signs. The conditions are inspired by Perron-Frobenius theory and are related to the algebraic connectivity of the network. We also show how all of these equilibria must be contained in a solid disk of radius given by the norm of the equilibrium point, which is located in the positive orthant.