Consensus and polarization in Altafini's model with bidirectional time-varying network topologies

The mechanism of reaching consensus in multi-agent systems has been exhaustively studied in recent years, motivated by numerous applications in engineering and science. Most consensus algorithms examined in the literature are based on the assumption about mutual trust and cooperation between the agents, implemented in the form of attractive couplings between the agents that render the values of the agents' states closer. However, in opinion dynamics of social groups, competition or antagonism between some pairs of agents is ubiquitous, which is usually characterized by the repulsive coupling, and may lead to clustering and polarization of opinions. A simple yet insightful model of opinion dynamics with antagonistic interactions was proposed recently by C. Altafini, which examined conventional first-order consensus algorithms with static signed interaction graphs, where the positive weight of an arc implies cooperation between the two agents and the negative one corresponds to antagonism. 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 time-varying and undirected. We give necessary and sufficient conditions under which the consensus protocol with the time-varying signed Laplacian establishes agreement of opinions in moduli, whose signs may be opposite, so that the agents' opinions either reach consensus or polarize.