On the Convergence Properties of Social Hegselmann-Krause Dynamics

TL;DR

This paper analyzes the convergence behavior of a variant of the Hegselmann-Krause opinion dynamics model that incorporates a physical connectivity graph, revealing conditions for infinite expected termination time and bounds on convergence rates.

Contribution

It introduces the social HK model with physical connectivity, establishes conditions for infinite expected convergence time, and characterizes the convergence properties based on graph structures.

Findings

Expected termination time can be infinite for connected, incomplete graphs.

Bounded $ ext{ extepsilon}$-convergence times are linked to specific graph properties.

Complete $r$-partite graphs have bounded $ ext{ extepsilon}$-convergence times.

Abstract

We study the convergence properties of Social Hegselmann-Krause dynamics, a {variant} of the Hegselmann-Krause (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 terminate 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 $\epsilon$-convergence time in terms of the properties of the physical connectivity graph. We extend this discussion and observe that for almost all $n$, there exists an $n$-vertex physical connectivity graph on which social HK dynamics may not even $\epsilon$-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 $\epsilon$-convergence to the steady state). Using the necessary conditions, we show that complete $r$-partite graphs have bounded $\epsilon$-convergence times.