Consensus and disagreement: the role of quantized behaviours in opinion dynamics
For researchers in opinion dynamics and multi-agent systems, this work identifies a fundamental limitation of quantized interactions, showing that consensus is not guaranteed and the deviation can be large.
This paper studies opinion dynamics where continuous opinions are influenced only by discrete behaviors, leading to discontinuous dynamics. The authors show that Carathéodory solutions can converge to non-consensus points arbitrarily far from consensus, and for Krasowskii solutions, the asymptotic distance from consensus is quadratic in the number of agents, indicating that quantization severely hinders consensus.
This paper deals with continuous-time opinion dynamics that feature the interplay of continuous opinions and discrete behaviours. In our model, the opinion of one individual is only influenced by the behaviours of fellow individuals. The key technical difficulty in the study of these dynamics is that the right-hand sides of the equations are discontinuous and thus their solutions must be intended in some generalized sense: in our analysis, we consider both Carathéodory and Krasowskii solutions. We first prove existence and completeness of Carathéodory solutions from every initial condition and we highlight a pathological behaviour of Carathéodory solutions, which can converge to points that are not (Carathéodory) equilibria. Notably, such points can be arbitrarily far from consensus and indeed simulations show that convergence to non-consensus configurations is very common. In order to cope with these pathological attractors, we then study Krasowskii solutions. We give an estimate of the asymptotic distance of all Krasowskii solutions from consensus and we prove its tightness via an example: this estimate is quadratic in the number of agents, implying that quantization can drastically destroy consensus. However, we are able to prove convergence to consensus in some special cases, namely when the communication among the individuals is described by either a complete or a complete bipartite graph.