A Bounded-Confidence Model of Opinion Dynamics on Hypergraphs

The paper proves consensus on the complete hypergraph for bounded initial opinions when N > ((b−a)/sqrt(c) + 1)((b−a)^2/c − 1) by combining (i) a bound on the number of limiting clusters (via a √(2c) spacing argument) and (ii) a cluster-size/discordance contradiction (Lemma 3.8), yielding Theorem 3.10 . The candidate solution establishes the same threshold by a constructive, probabilistic argument: it finds a large concordant hyperedge inside a √c-interval, creates a large equal-valued cluster, and then shows it almost surely absorbs all outsiders in finite time under uniform i.i.d. hyperedge sampling (consistent with the model’s update rule and choice d = d1) . The model’s proof is sound provided one states explicitly that the hypergraph is complete and c>0; its “success-run” argument even yields finite-time consensus, which is compatible with the paper’s general finite-time convergence result for complete hypergraphs that contain all prime-sized subsets . Therefore, both are correct, with different proofs.