Consensus and disagreement: the role of quantized behaviours in opinion dynamics

TL;DR

This paper investigates how quantized behaviors influence opinion dynamics, revealing that quantization can hinder consensus and exploring solution types to understand these effects.

Contribution

It introduces a model combining continuous opinions with discrete behaviors, analyzing solution types and their impact on consensus in opinion dynamics.

Findings

Carathéodory solutions can converge to non-equilibrium points far from consensus.

Krasowskii solutions provide bounds on the distance from consensus, showing quantization's disruptive effect.

Consensus can be achieved in special cases like complete or bipartite graphs.

Abstract

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 Carath\'eodory and Krasowskii solutions. We first prove existence and completeness of Carath\'eodory solutions from every initial condition and we highlight a pathological behaviour of Carath\'eodory solutions, which can converge to points that are not (Carath\'eodory) 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.