On the Convergence Properties of Social Hegselmann-Krause Dynamics

• Published in: IEEE transactions on automatic control Vol. 67; no. 2; pp. 589 - 604
• Main Authors: Parasnis, Rohit Yashodhar, Franceschetti, Massimo, Touri, Behrouz
• Format: Journal Article
• Language: English
• Published: New York IEEE 01.02.2022; The Institute of Electrical and Electronics Engineers, Inc. (IEEE)
• Subjects: Connectivity; Context modeling; Convergence; Dynamics; Eigenvalues and eigenfunctions; Graph theory; Laplace equations; Merging; Multiagent systems; network theory (graphs); networked control systems; nonlinear systems; Steady state; Upper bound
• ISSN: 0018-9286; 1558-2523
• DOI: 10.1109/TAC.2021.3052748

We study the convergence properties of social Hegselmann-Krause (HK) dynamics , a variant of the HK model of opinion dynamics where a physical connectivity graph that accounts for the extrinsic factors that could prevent interaction between certain pairs of agents is incorporated. As opposed to the original HK dynamics (which terminates in finite time), we show that for any underlying connected and incomplete graph, under a certain mild assumption, the expected termination time of social HK dynamics is infinity. We then investigate the rate of convergence to the steady state, and provide bounds on the maximum <inline-formula><tex-math notation="LaTeX">\epsilon</tex-math></inline-formula>-convergence time in terms of the properties of the physical connectivity graph. We extend this discussion and observe that for almost all <inline-formula><tex-math notation="LaTeX">n</tex-math></inline-formula>, there exists an <inline-formula><tex-math notation="LaTeX">n</tex-math></inline-formula>-vertex physical connectivity graph on which social HK dynamics may not even <inline-formula><tex-math notation="LaTeX">\epsilon</tex-math></inline-formula>-converge to the steady state within a bounded time frame. We then provide nearly tight necessary and sufficient conditions for arbitrarily slow merging (a phenomenon that is essential for arbitrarily slow <inline-formula><tex-math notation="LaTeX">\epsilon</tex-math></inline-formula>-convergence to the steady state). Using the necessary conditions, we show that complete <inline-formula><tex-math notation="LaTeX">r</tex-math></inline-formula>-partite graphs have bounded <inline-formula><tex-math notation="LaTeX">\epsilon</tex-math></inline-formula>-convergence times.