Holme–Newman co-evolution phase transition

Establish whether the Holme–Newman co-evolving network–opinion model—where at each time step a uniformly chosen vertex rewires one incident edge to a vertex with the same opinion with probability β, or adopts the opinion of a uniformly chosen neighbor with probability 1−β—exhibits a phase transition between consensus and polarisation as the parameter β varies, as suggested by simulation evidence.

Background

The paper reviews co-evolutionary voter models on dense graphs, where both opinions and edges evolve. In related models studied by Durrett et al. and by Basu and Sly, rigorous results show distinct regimes for the absorption time and the final fraction of minority opinions, suggesting a transition in behavior controlled by a parameter β.

Holme and Newman proposed a different update scheme that alternates opinion adoption and rewiring-to-same depending on β; the review notes their conjecture, based on simulations, of a phase transition similar to that predicted in earlier work, making the rigorous existence of such a transition for their specific scheme an explicit open conjecture highlighted by the author.

References

Holme and Newman conjectured a similar phase transition as that conjectured in , based on simulation.

— Evolution of Discordance  (2410.17808 - Hollander, 2024) in Subsection 3.1 (Attempts at capturing co-evolution)

Although we can't argue for the discontinuous nature of this transition without a finite-size analysis, which is excluded in the case of a specific empirical network, the observed decrease of the transition region for fuzzy neighbor selection is important for understanding the evolution of opinions in real settings, such as online social networks.

— Probabilistic neighbors' selection competes with confirmation bias in a bounded confidence model  (2610.07971 - Giaquinta et al., 6 Oct 2026) in Discussion, Section 4