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

In 2012, Mirtabatabaei and Bullo studied heterogeneous Hegselmann–Krause models of opinion dynamics, where different agents may have different confidence or influence bounds. They conjectured that opinion vectors always converge, a fundamental problem that remains open. In support of this main conjecture they gave some partial results and made some auxiliary conjectures. Here we give counterexamples to two of these auxiliary conjectures, and even to one theorem in their paper. We explain why these disproved conjectures and false theorem nevertheless likely remain valid for almost all initial opinion vectors. In the last section (Section 7), we describe the role of AI in the production of the results in this paper.