Source-linked AI summary
Higher-order interactions shape collective dynamics differently in hypergraphs and simplicial complexes
Yuanzhao Zhang, Maxime Lucas, Federico Battiston
TL;DR
The paper examines whether choosing a hypergraph or simplicial-complex representation changes conclusions about higher-order collective dynamics. Using phase-oscillator synchronization and theoretical analysis of degree heterogeneity and cross-order degree correlations, it finds opposite effects of higher-order interactions across representations. The results indicate that these representations should not always be treated as interchangeable.
Problem
The paper asks whether the commonly convenience-driven choice between hypergraphs and simplicial complexes has hidden consequences for collective dynamics.
Method
The authors study synchronization in identical phase oscillators on random, structured, and brain-derived hypergraphs and simplicial complexes, linking stability to higher-order degree heterogeneity and cross-order degree correlations.
Results
Higher-order interactions tend to stabilize synchronization in broad classes of hypergraphs but destabilize it in simplicial complexes; in random hypergraphs, mixed pairwise and non-pairwise coupling can produce an optimal intermediate α.
Takeaways & Limitations
Hypergraphs and simplicial complexes cannot always be used interchangeably when interpreting collective dynamics with nonpairwise interactions.
Abstract
from arXiv · showhide
Higher-order networks have emerged as a powerful framework to model complex systems and their collective behavior. Going beyond pairwise interactions, they encode structured relations among arbitrary numbers of units through representations such as simplicial complexes and hypergraphs. So far, the choice between simplicial complexes and hypergraphs has often been motivated by technical convenience. Here, using synchronization as an example, we demonstrate that the effects of higher-order interactions are highly representation-dependent. In particular, higher-order interactions typically enhance synchronization in hypergraphs but have the opposite effect in simplicial complexes. We provide theoretical insight by linking the synchronizability of different hypergraph structures to (generalized) degree heterogeneity and cross-order degree correlation, which in turn influence a wide range of dynamical processes from contagion to diffusion. Our findings reveal the hidden impact of higher-order representations on collective dynamics, highlighting the importance of choosing appropriate representations when studying systems with nonpairwise interactions.
I. INTRODUCTION
Higher-order interactions are common in complex systems, but their representation in hypergraphs or simplicial complexes has often been chosen for technical convenience. The paper asks whether this choice has hidden consequences for collective dynamics, using synchronization to investigate when higher-order interactions promote or impede synchrony.
- Motivation: Traditional networks encode only pairwise interactions, whereas many systems involve nonlinear influences among multiple units that cannot be decomposed into pairs.Examples span human dynamics, collaborations, ecological systems, and the brain.
- Representation gap: Hypergraphs and simplicial complexes are commonly used to represent higher-order interactions, and the two have often been treated as interchangeable.The choice is frequently motivated by technical convenience or by the requirements of particular methods.
- Data constraint: Reliable real-world hypergraph data remain scarce, so higher-order connections are often inferred from pairwise networks.A common homophily-based practice attaches three-body interactions to closed triangles, effectively selecting a simplicial-complex representation.
- Focus: Synchronization provides a paradigmatic setting for testing when higher-order interactions promote collective order.It underlies natural and engineered systems, and nonpairwise interactions arise naturally from phase reduction of coupled oscillator populations.
- Central finding: The paper shows that the effect of higher-order interactions is representation-dependent: they destabilize synchronization in simplicial complexes but tend to stabilize it in broad classes of hypergraphs.The proposed explanation links these opposite trends to higher-order degree heterogeneity and cross-order degree correlations.
A. Higher-order interactions hinder synchronization in simplicial complexes but facilitate it in random hypergraphs
The study models identical phase oscillators with pairwise and three-body couplings, then evaluates synchronization stability through the second Lyapunov exponent. Simplicial complexes and random hypergraphs show opposite responses to higher-order coupling, while mixed orders can optimize synchronization in random hypergraphs.
- Model: The model extends the Kuramoto framework to unweighted, undirected interactions up to order two, including three-body coupling.Adjacency tensors identify pairwise and second-order interactions, with coupling strengths γ1 and γ2.
- Model: The parameter α shifts the coupling from entirely first-order at α = 0 to entirely second-order at α = 1 while keeping the total coupling budget constant.Each order’s coupling is also normalized by its average degree.
- Stability measure: Synchronization stability is determined by λ2: λ2 < 0 indicates stable synchrony, and larger absolute values indicate faster recovery from perturbations.λ2 is obtained from the eigenvalues of the multiorder Laplacian after linearization around the synchronized state.
- Structures: Random hypergraphs are generated independently by wiring probabilities p1 = p and p2 = p△, whereas simplicial complexes add a three-body interaction to every triangle in an Erdős–Rényi graph.The simplicial-complex construction imposes that every higher-order interaction includes all corresponding lower-order interactions.
- Main result: Higher-order interactions impede synchronization in simplicial complexes but improve it in random hypergraphs; for random hypergraphs with p significantly larger than p△, λ2(α) becomes U-shaped with an optimum 0 < α* < 1.The simplicial-complex trend holds for all p, while the opposite monotonic trend occurs for random hypergraphs when p ≃ p△.
- Robustness: The observed random-hypergraph behavior persists under equalized connection counts and when simplicial complexes are formed by probabilistically filling triangles, provided the filling probability is not too close to zero.Similar results are also reported for simplicial complexes built from small-world and scale-free networks.
B. Linking higher-order representation, degree heterogeneity, and synchronization performance
Degree heterogeneity explains why higher-order interactions have opposite synchronization effects in simplicial complexes and random hypergraphs. In simplicial complexes, higher-order degrees amplify heterogeneity, whereas in random hypergraphs they reduce relative heterogeneity and narrow the Laplacian spectrum.
- Degree heterogeneity and stability: Degree-based bounds connect Laplacian spectral extremes, and therefore synchronization stability, to minimum and maximum generalized degrees.The bounds relate λn to maximum degree and λ2 to minimum degree.
- Simplicial complexes: In simplicial complexes, second-order degrees depend quadratically on first-order degrees, producing a rich-get-richer effect.Well-connected nodes and regions receive disproportionately more higher-order coupling.
- Simplicial complexes: r > 1 means 2-simplices are more degree-heterogeneous than the pairwise network, predicting worse synchronization stability with higher-order interactions.For Erdős–Rényi-derived simplicial complexes, r remains above one and grows as pairwise connections become sparser.
- Random hypergraphs: In random hypergraphs, second-order degree heterogeneity is lower than first-order heterogeneity, with the difference increasing as system size n grows.The theoretical lower bound agrees well with simulations using p = p△ = 0.1.
- Random hypergraphs: The narrower normalized second-order Laplacian spectrum explains why higher-order interactions improve synchronization stability in random hypergraphs.Larger combinatorial degree scales concentrate binomial degrees around their means, reducing relative fluctuations.
C. Exploring the hypergraph space with synthetic networks and brain networks
Moving hypergraph structures away from simplicial complexes causes higher-order interactions to switch rapidly from impeding synchronization to promoting it. This pattern holds across brain-derived, synthetic, and other real-world hypergraphs.
- Higher-order interactions quickly switch from impeding to promoting synchronization as hypergraphs move farther from simplicial complexes.
- In brain-derived hypergraphs, synchronization stability consistently improves as the structures move farther from simplicial complexes.
- For all six brain networks, increasing shuffling probability changes synchronization curves from upward or level to downward with α.The curves were averaged over independent hypergraph realizations at each shuffling probability.
- At ps = 0, higher-order interactions impede synchronization; as ps increases, all six systems rapidly enter the synchronization-promoting region.The transition is represented by crossing the diagonal in the λ2(α = 0.5) versus λ2(α = 0) plot.
- The same transition appears in hypergraphs built from synthetic networks and real-world hypergraphs, provided network density is not too sparse or dense.Changing density mainly shifts curves vertically without affecting the transition.
D. The role of degree correlation
Cross-order degree correlation provides a distinct structural factor affecting synchronization when pairwise and nonpairwise interactions coexist. Reducing this correlation can improve stability by allowing degree heterogeneity across interaction orders to compensate.
- Cross-order degree correlation is large and positive in simplicial complexes but close to zero in random hypergraphs.
- The membership-swap method lowers degree correlation while preserving degree heterogeneity by exchanging 2-simplex memberships between nodes.Iteratively swapping nodes with the lowest and largest k(2) maximizes the correlation change.
- In the cat brain network, swapping 5 or 15 node pairs lowers λ2 at intermediate α while leaving λ2(α = 0) and λ2(α = 1) unchanged.The endpoint invariance follows because only one interaction order is present at each endpoint.
- Lowering cross-order degree correlation can improve synchronization stability when pairwise and nonpairwise interactions are mixed.
- Negative cross-order correlation can homogenize hypergraph structure by compensating degree heterogeneity across interaction orders.
III. DISCUSSION
Higher-order interactions promote synchronization in many hypergraphs but impede it in simplicial complexes. These opposite trends are linked to higher-order degree heterogeneity and degree correlation, with implications beyond synchronization.
- Higher-order interactions promote synchronization in a broad class of hypergraphs but impede it in simplicial complexes.
- Higher-order degree heterogeneity and degree correlation underlie the opposite synchronization trends.
- The framework considered two-body and three-body couplings but naturally extends to larger group interactions.
- Generalized Laplacians and their eigenvalue spreads suggest that the findings may transfer to generic oscillator dynamics after normalizing coupling functions.
- The results may also bear on diffusion, contagion, and evolutionary processes where degree heterogeneity and correlation matter.
- The findings suggest that simplicial complexes and hypergraphs cannot always be used interchangeably when interpreting collective dynamics.
Supplementary Information
The supplementary analyses test whether representation-dependent synchronization patterns persist across random, scale-free, small-world, and real-world structures. They also examine robustness to triangle filling, network structure, and cross-order degree correlation.
- Random hypergraphs enhance synchronization, whereas simplicial complexes impede it under matched first- and second-order interaction counts.
- Simplicial complexes impede synchronization across varying probabilities of closed-triangle filling.
- Simplicial complexes constructed from scale-free networks show that nonpairwise interactions impede synchronization.
- Simplicial complexes constructed from small-world networks likewise show impaired synchronization under nonpairwise interactions.
- Hypergraphs built from scale-free and small-world networks generally show enhanced synchronization, except when their structure approaches a simplicial complex.
- Real-world hypergraphs from hospital and school social contacts are analyzed despite being close to simplicial complexes.
- More negative cross-order degree correlation improves synchronization stability in mixed-order systems, while uniform cases are unaffected.