Convergence of heterogeneous Hegselmann–Krause opinion dynamics

Prove that, for every finite heterogeneous Hegselmann–Krause system in either the synchronized bounded confidence (SBC) model or the synchronized bounded influence (SBI) model, the opinion vector converges in the Euclidean topology as time tends to infinity.

Background

The paper studies heterogeneous Hegselmann–Krause opinion dynamics in which each agent has an individual positive confidence or influence bound. In the SBC model, agent i averages the opinions within its own confidence radius, whereas in the SBI model, agent i averages the opinions of agents whose individual influence radii reach agent i.

Mirtabatabaei and Bullo formulated the convergence assertion as Conjecture 2.1. The paper disproves several auxiliary conjectures and a related theorem concerning finite-time stabilization of the proximity digraph and pseudostable convergence, but it does not resolve the broader question of whether every opinion vector in the heterogeneous SBC or SBI models converges.

References

Mirtabatabaei and Bullo conjectured (Conjecture 2.1 in ) that, for any initial opinion vector ${x}(0) \in \mathbb{R}n$, the vectors ${x}(t)$ converge (in the usual Euclidean topology, thus pointwise) as $t \rightarrow \infty$. This conjecture remains open and is definitely one of the outstanding open problems concerning Hegselmann--Krause dynamics.

— Non-freezing of the proximity digraph and non-pseudostable convergence in heterogeneous Hegselmann--Krause models: Counterexamples to conjectures and results of Mirtabatabaei and Bullo  (2610.03229 - Hegarty et al., 2 Oct 2026) in Section 1, Introduction