Closed theory for Kuramoto dynamics with nested hyperedges

Develop a closed analytical theory for Kuramoto synchronization with pairwise and three-body interactions on regular hypergraphs featuring tunable inter-order hyperedge overlap α, capturing the effect of nested hyperedges and providing explicit analytical conditions for the synchronization onset σ1* and for the bistability threshold \hat{σ2}.

Background

The main text develops a microscopic theory for SIS contagion on hypergraphs with tunable inter-order nestedness and provides analytical conditions for the onset and nature of the transition, validated by simulations. To assess generality, the authors study higher-order Kuramoto synchronization on the same class of hypergraphs but rely on numerical exploration because they lack a closed analytic framework in this setting.

They observe that nested hyperedges qualitatively anticipate synchronization onset and suppress explosive transitions, mirroring the SIS phenomenology. However, without a closed theory, the onset and bistability thresholds for Kuramoto dynamics are extracted numerically rather than derived analytically, leaving a clear theoretical gap to formalize the role of inter-order overlap in synchronization.

References

Unlike the SIS case, where our mean-field theory and center-manifold analysis provide analytical conditions for both the onset and the bistability threshold, here we do not have a closed theory capturing the effect of nested hyperedges on Kuramoto dynamics; therefore, both quantities are extracted numerically.

— Nested hyperedges promote the onset of collective transitions but suppress explosive behavior  (2601.10522 - Malizia et al., 15 Jan 2026) in Supplementary Section I: Effects of nestedness on Kuramoto dynamics