Nonmonotonic consensus transitions in bounded-confidence dynamics on unbiased networks

Abstract: We study the Hegselmann-Krause model of opinion dynamics on sparse, unbiased networks generated via Wilson's algorithm, unveiling how network connectivity and confidence bounds jointly determine collective behavior. By systematically exploring the parameter space spanned by the confidence level $\epsilon$ and the mean degree density $\mu$, we construct comprehensive phase diagrams that classify the emergent steady states into different degrees of fragmentation and consensus. We uncover a nonmonotonic re-entrant transition where increased connectivity can paradoxically suppress consensus, and show that full unanimity is unattainable at low connectivity due to structural isolation. Convergence times exhibit two distinct slowdowns: a finite-size, connectivity-dependent resonance near $\epsilon \sim 1/N$, and a critical peak associated with the established fragmentation-to-consensus transition. While the critical confidence threshold $\epsilon_c$ stabilizes near 0.2 for large system sizes, finite-size effects and sparse connectivity significantly alter the dynamics and phase boundaries in smaller populations. Our results offer new insights into the interplay between network topology and opinion dynamics, and highlight conditions under which increased connectivity may hinder, rather than promote, consensus.