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

Abstract

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.

arXiv commentVersion 2. In V.1, the conditions under which Thm 6.4(iii)(a) of [4] can be rescued were not described accurately. Minor editions to Abstract, Sections 1 and 6. Since V.1 appeared, our results have notably been built on. Here we follow the conventional practice, by describing these developments in a note at the end of the paper, but otherwise retaining the structure of the original submission