Opinion dynamics using Altafini´s model with a time-varying directed graph

Distributed control policies (or protocols) for multi-agent consensus have been extensively studied in recent years, motivated by numerous applications in engineering and science. Most of these algorithms assume the agents to be mutually cooperative and “trustful” and correspondingly attractive couplings between the agents bring the values of the agents´ states closer. Opinion dynamics of real social groups, however, require beyond conventional models of multi-agent consensus due to ubiquitous competition and distrust between some pairs of agents, which are usually characterized by the repulsive cou-pling. Antagonistic interactions prevent the averaging tendency of the opinions, which cooperative consensus protocols promote, and may lead to their polarization and clustering. A simple yet insightful model of opinion dynamics with both attractive and repulsive couplings was proposed recently by C. Altafini, who examined first-order consensus algorithms over static signed interaction graphs, where arcs of positive weights connect cooperating agents, and of negative weights correspond to antagonistic pairs. This protocol establishes modulus consensus, where the opinions become the same in modulus but may differ in sign. In the present paper, we extend the modulus consensus model to the case where the network topology is arbitrary time-varying, directed, signed graph. We show that under mild condition of uniform strong connectivity of the network, the protocol establishes agreement of opinions in moduli, whose signs may be opposite, so that the agents´ opinions either reach consensus or polarize. This result is further extended to nonlinear consensus protocols. We show also that, unlike cooperative consensus algorithms, uniform strong connectivity cannot be relaxed to uniform quasi-strong connectivity (UQSC).