Nested hyperedges promote the onset of collective transitions but suppress explosive behavior

TL;DR

This work addresses how higher-order interactions organized as nested hyperedges influence critical transitions in complex systems. The authors develop a microscopic mean-field theory for SIS dynamics on hypergraphs with 1- and 2-hyperedges and tunable inter-order overlap, introducing $α$ and rescaled infectivities $λ_1$ and $λ_2$. They find that increasing overlap $α$ anticipates the epidemic onset by reallocating transmission to internal group routes while simultaneously suppressing nonlinear feedback necessary for bistability, causing backward bifurcations and explosive transitions to shrink or vanish at a critical $α_c$. The phenomenon generalizes to higher-order synchronization (Kuramoto dynamics), suggesting a universal mechanism by which nested higher-order structure governs the emergence and nature of collective transitions with potential implications for controlling cascades in real systems.

Abstract

Higher-order interactions can dramatically reshape collective dynamics, yet how their microscopic organization controls macroscopic critical behavior remains unclear. Here we develop a new theory to study contagion dynamics on hypergraphs and show that nested hyperedges not only facilitate the onset of spreading, but also suppress backward bifurcations, thereby inhibiting explosive behavior. By disentangling contagion pathways, we find that overlap redirects transmission from external links to internal, group-embedded routes -- boosting early activation but making dyadic and triadic channels increasingly redundant. This loss of structural independence quenches the nonlinear amplification required for bistability, progressively smoothing the transition as hyperedges become nested. We observe the same phenomenology in Kuramoto dynamics, pointing to a broadly universal mechanism by which nested higher-order structure governs critical transitions in complex systems.

Figures (4)

• Figure 1: Higher-order contagion with nested hyperedges. (a) Schematic of a 2-hyperedge in state $\rho^{\rm ISI_\Delta}$ (two infected and one susceptible node). The inter-order overlap $\alpha$ weights the contribution of pairwise infections occurring on links embedded within 2-hyperedges (internal channel), relative to infections occurring on external links. (b) Illustration of the local neighborhood of a susceptible node participating in $k_2$ 2-hyperedges. Increasing overlap increases the number of pairwise contacts that are internal to the groups and reduces the number of external links available for transmission. (c) Example configuration in which a susceptible node is shared by two 2-hyperedges, one of which contains two infected nodes, highlighting how overlap couples dyadic and triadic transmission pathways.
• Figure 2: Nested hyperedges promote the onset of collective transitions but suppress explosive behavior. (a) Nonlinear coefficient $a$ from center-manifold theory as a function of overlap $\alpha$, for $\lambda_2\in\{1,2,3,4\}$ (with $k_1=5$, $k_2=2$). Vertical dashed lines mark the critical overlap $\alpha_c$ at which $h=0$ and the transition changes from subcritical (bistable) to supercritical (continuous). (b) Critical three-body infectivity $\hat{\lambda}_2$ (solid) required for bistability and the corresponding epidemic threshold $\lambda_1^*$ (dashed), both obtained numerically as functions of $\alpha$ for $k_1=5$, $k_2=2$. (c) Phase diagram in the $(\alpha,\lambda_1)$ plane predicted by the model for $k_1=5$, $k_2=2$, and $\lambda_2=3$, showing that increasing $\alpha$ lowers $\lambda_1^*$ while shrinking the bistable region, which disappears for $\alpha>\alpha_c$, yielding continuous transitions. (d) Stationary infected density $\rho^*$ from theory (lines) and Gillespie simulations (markers) on random regular hypergraphs with $N=3000$, $k_1=5$, and $k_2=2$, for three representative values of $\alpha$ at $\lambda_2=3$. For $\alpha=1$, forward and backward branches coincide (continuous transition); for $\alpha=0.5$, bistability emerges (forward: solid/blue; backward: dashed/red); for $\alpha=0$, bistability is maximal and the forward threshold occurs at $\lambda_1^{*,(0)}$ (vertical dotted lines).
• Figure 3: Microscopic mechanisms underlying the anticipated onset and suppressed bistability. (a) Quasi-stationary states for the fast variables $\Pi$ and $\delta$ as functions of the inter-order overlap $\alpha$, obtained from theory (lines) and Gillespie simulations (symbols) on random regular hypergraphs with $N=3000$, $k_1=5$, $k_2=2$, and $\lambda_2=3$. Each point is evaluated close to $\lambda_1^*$ for each $\alpha$. The only nonvanishing contribution to $\bar{\delta}$ arises from internal pairwise contagion within groups and increases with $\alpha$. (b,c) Decomposition of the stationary infected density $\rho^*$ into total, pairwise (1-hyperedge), and group-based (2-hyperedge) contributions for $\alpha=0$ and $\alpha=1$, evaluated at the corresponding critical values $\hat{\lambda}_2$, before bistability emerges. In both cases the transition remains continuous, although nested structures display a sharper onset. (d) Fractions of accumulated infection events $\eta_1^*$ (pairwise, internal and external) and $\eta_2^*$ (higher-order) near $\lambda_1^*$ as functions of $\alpha$, showing a progressive shift from external pairwise to internal and higher-order transmission channels. (e) Critical group infectivity $\hat{\lambda}_2$ versus $k_1$ (with $k_2=2$) for $\alpha=0$ (dashed, analytical) and $\alpha=1$ (solid, numerical), illustrating the strong suppression of explosive behavior induced by overlap in sparse networks; both curves converge to the mean-field limit $\hat{\lambda}_2=1$ (dotted line) for large $k_1$.
• Figure S1: Nested hyperedges shape synchronization onset and explosive behavior in Kuramoto dynamics. (a) Critical pairwise coupling $\sigma_1^*$ (circles) and critical three-body coupling $\hat{\sigma}_2$ (filled symbols) as functions of the inter-order overlap $\alpha$, obtained numerically from simulations on regular hypergraphs with $N=300$, $k_1=5$, and $k_2=2$. (b--d) Time-averaged Kuramoto order parameter $\langle r \rangle$ as a function of $\sigma_1$ for $\sigma_2=3$ and $\alpha=1$, $0.5$, and $0$, respectively. Solid (dashed) lines correspond to forward (backward) continuation. Increasing nestedness anticipates the synchronization onset while progressively suppressing bistability and explosive synchronization.