Continuous-time average-preserving opinion dynamics with opinion-dependent communications

Blondel, Vincent;Hendrickx, Julien;Tsitsiklis, John
(2010) SIAM Journal on Control and Optimization — Vol. 48, n° 8, p. 5214-5240 (2010)

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Abstract
We study a simple continuous-time multi-agent system related to Krause’s model of opinion dynamics: each agent holds a real value, and this value is continuously attracted by every other value differing from it by less than 1, with an intensity proportional to the difference. We prove convergence to a set of clusters, with the agents in each cluster sharing a common value, and provide a lower bound on the distance between clusters at a stable equilibrium, under a suitable notion of multi-agent system stability. To better understand the behavior of the system for a large number of agents, we introduce a variant involving a continuum of agents. We prove, under some conditions, the existence of a solution to the system dynamics, convergence to clusters, and a non-trivial lower bound on the distance between clusters. Finally, we establish that the continuum model accurately represents the asymptotic behavior of a system with a finite but large number of agents.
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Blondel, V., Hendrickx, J., & Tsitsiklis, J. (2010). Continuous-time average-preserving opinion dynamics with opinion-dependent communications. SIAM Journal on Control and Optimization, 48(8), 5214-5240. https://doi.org/10.1137/090766188 (Original work published 2010)