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  • Open Access

Nonmonotonic consensus transitions in bounded-confidence dynamics on unbiased networks

Paolo Molignini*

  • Department of Physics and Nanoscience Center, University of Jyväskylä, P.O. Box 35 (YFL), University of Jyväskylä, FI-40014 Jyväskylä, Finland and Department of Physics, Stockholm University, AlbaNova University Center, 106 91 Stockholm, Sweden

  • *Contact author: paolo.molignini@fysik.su.se
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Phys. Rev. E 113, 024304 – Published 9 February, 2026

DOI: https://doi.org/10.1103/y2kw-b7n5

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 ε and the mean degree density μ, we construct comprehensive phase diagrams that classify the emergent steady states into different degrees of fragmentation and consensus. We uncover a nonmonotonic reentrant 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 ε∼1/N, and a critical peak associated with the established fragmentation-to-consensus transition. While the critical confidence threshold εc 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 insights into the interplay between network topology and opinion dynamics, and highlight conditions under which increased connectivity may hinder, rather than promote, consensus.

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Phys. Rev. E 113, 024304– Published 9 February, 2026
  • Received 4 August 2025
  • Accepted 15 January 2026
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Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI. Funded by Bibsam.

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