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 Krasovskii solutions. We first prove existence and completeness of Carath'e-o-dory solutions from every initial condition and we highlight a pathological behavior of Carath'eo-do-ry 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 Krasovskii solutions. We give an estimate of the asymptotic distance of all Krasovskii solutions from consensus and we prove its tightness by an example of equilibrium such that this distance is quadratic in the number of agents. This fact implies that quantization can drastically destroy consensus. However, consensus is guaranteed in some special cases, namely when the communication among the individuals is described by either a complete or a complete bipartite graph.

Discontinuities, Generalized Solutions, and (Dis)agreement in Opinion Dynamics / Ceragioli, F.; Frasca, P. - In: Control Subject to Computational and Communication ConstraintsSTAMPA. - [s.l] : Springer, 2018. - ISBN 978-3-319-78449-6. - pp. 287-309

Discontinuities, Generalized Solutions, and (Dis)agreement in Opinion Dynamics

F. Ceragioli;
2018

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 Krasovskii solutions. We first prove existence and completeness of Carath'e-o-dory solutions from every initial condition and we highlight a pathological behavior of Carath'eo-do-ry 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 Krasovskii solutions. We give an estimate of the asymptotic distance of all Krasovskii solutions from consensus and we prove its tightness by an example of equilibrium such that this distance is quadratic in the number of agents. This fact implies that quantization can drastically destroy consensus. However, consensus is guaranteed in some special cases, namely when the communication among the individuals is described by either a complete or a complete bipartite graph.
978-3-319-78449-6
Control Subject to Computational and Communication Constraints
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/11583/2709339