Graph stabilization and pseudostability in heterogeneous Hegselmann–Krause dynamics

In synchronized heterogeneous Hegselmann–Krause bounded-confidence and bounded-influence opinion models, ask whether proximity digraphs eventually stabilize and whether constant-digraph trajectories eventually become pseudostable. Convergence of every trajectory is a separate conjecture.

Why it matters and what counts as progress

Why it matters: It separates convergence of opinions from eventual stability of the interaction graph and from pseudostability.

What counts as progress: An exact trajectory disproving graph stabilization or pseudostability, or a theorem characterizing when those properties hold. Trajectory convergence is a separate question.

How the problem developed

2012 · Conjecture [1]
Anahita Mirtabatabaei and Francesco Bullo Studied heterogeneous bounded-confidence and influence models, proved convergence under sufficient conditions, and conjectured eventual convergence to a steady state with fixed topology.
2 Oct 2026 · Counterexamples [2]
Peter Hegarty, Damiano Ognissanti, Edvin Wedin, and Claude Opus 5.5 Gave counterexamples to two auxiliary conjectures and one theorem from the 2012 paper; the main opinion-vector convergence conjecture remains open.

AI contribution

Non-freezing of the proximity digraph and non-pseudostable convergence in heterogeneous Hegselmann--Krause models: Counterexamples to conjectures and results of Mirtabatabaei and Bullo

Peter Hegarty, Damiano Ognissanti, Edvin Wedin

Hegarty, Ognissanti and Wedin give convergent heterogeneous Hegselmann–Krause trajectories in the synchronized bounded-confidence and bounded-influence models whose proximity digraphs never stabilize, plus constant-digraph trajectories that fail eventual pseudostability. These counterexamples refute Mirtabatabaei and Bullo Conjectures 2.2 and 2.3 and Theorem 6.4(iv).

AI involvementAI-led

ImportanceNotable

Technical verificationFormal (Lean 4.35.0-rc2/Mathlib + human)

Human understandingFull