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Interplay of hypergraph structure and thresholds in determining criticality

Characterize how hypergraph structural features (e.g., degree and cardinality distributions and their heterogeneity) jointly with the contagion and annihilation thresholds determine the existence, location, and nature of the critical point and phase transitions in the threshold rumor model on hypergraphs.

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Background

The authors find that the critical point and phase transition region depend non-trivially on both the hypergraph’s structure and the threshold parameters, noting empirical differences between power-law and truncated Poisson hypergraphs. However, a quantitative theory explaining this interplay is lacking.

Developing an analytical understanding of the combined influence of structure and thresholds on critical phenomena in this model is explicitly identified as an open problem.

References

Despite the nature of the transition, the critical point changes as a function of both the structure and the contagion and annihilation thresholds. The interplay between these quantities is not trivial. Indeed, we find that the region of the parameter space $\Theta_\lambda \times \Theta_\alpha$ in which a phase transition is observed is significantly smaller for power-law hypergraphs than for the truncated Poisson hypergraphs (see Appendix~\ref{apdx:critical_point_grid} for additional experiments). Understanding the interplay between the structure and the contagion and annihilation thresholds is another open problem.

— Rumor propagation on hypergraphs (2504.19305 - Oliveira et al., 27 Apr 2025) in Discussion and Conclusions, Subsection "Critical behavior" (Section 6.3)