Collective dynamics on higher-order networks (2510.05253v1)

Abstract: Higher-order interactions that nonlinearly couple more than two nodes are ubiquitous in networked systems. Here we provide an overview of the rapidly growing field of dynamical systems with higher-order interactions, and of the techniques which can be used to describe and analyze them. We focus in particular on new phenomena that emerge when nonpairwise interactions are considered. We conclude by discussing open questions and promising future directions on the collective dynamics of higher-order networks.

• The paper presents a generalized Kuramoto model that incorporates both pairwise and nonpairwise interactions via adjacency tensors to unveil their unique contributions to synchronization.
• It reveals that multiorder Laplacians govern the stability of synchrony and expose contrasting effects in hypergraphs versus simplicial complexes.
• Analysis of nonlinear dynamics demonstrates that higher-order interactions induce explosive transitions, hysteresis, and extensive multistability in oscillator ensembles.

Collective Dynamics on Higher-Order Networks: A Technical Review

Introduction and Motivation

The paper of collective dynamics in networked systems has traditionally focused on pairwise interactions, as exemplified by the classical Kuramoto model. However, empirical evidence from neuroscience, ecology, and physics increasingly points to the ubiquity and importance of higher-order (nonpairwise, polyadic) interactions, which are more accurately modeled by hypergraphs and simplicial complexes. This review synthesizes recent advances in the theory and analysis of dynamical systems with higher-order interactions, emphasizing the emergence of novel phenomena such as explosive synchronization transitions, multistability, and complex collective behaviors that are unattainable in pairwise frameworks.

Figure 1: Coupled dynamical units (e.g., Kuramoto oscillators) exhibit ordered/disordered states; higher-order structure and coupling fundamentally alter the transition from incoherence to synchronization.

Kuramoto Oscillators with Higher-Order Interactions

Generalized Model Formulation

The generalized Kuramoto model incorporates both pairwise and nonpairwise (e.g., triadic) interactions via adjacency tensors, leading to equations of the form:

$$\dot{\theta}_i = \omega_i + \sigma \sum_{j=1}^{n} A_{ij} \sin(\theta_j-\theta_i) + \sigma_\triangle^{(\rm s)} \sum_{j,k=1}^{n} B_{ijk} \sin(\theta_j+\theta_k-2\theta_i) + \sigma_\triangle^{(\rm as)} \sum_{j,k=1}^{n} C_{ijk} \sin(2\theta_j-\theta_k-\theta_i)$$

This formulation enables the paper of how higher-order coupling strengths and network structure influence synchronization dynamics, stability, and the emergence of new collective states.

Stability Analysis and Multiorder Laplacians

For identical oscillators, the stability of full synchrony is determined by the spectrum of the multiorder Laplacian, which generalizes the graph Laplacian to incorporate higher-order adjacency tensors. The multiorder Laplacian aggregates the effects of pairwise and nonpairwise interactions, providing a unified framework for linear stability analysis.

Figure 2: Structural properties such as degree homogeneity, cross-order degree correlation, and intra-order overlap critically affect synchronization dynamics; simpliciality can either stabilize or destabilize synchrony depending on network representation.

A key result is the contradictory effect of higher-order interactions in hypergraphs versus simplicial complexes: higher-order interactions enhance synchronization in generic hypergraphs but impede it in simplicial complexes due to structural heterogeneity induced by downward closure.

Emergent Phenomena: Explosive Transitions and Multistability

In heterogeneous oscillator ensembles, higher-order interactions induce nonlinearities in the macroscopic order parameter dynamics, leading to subcritical bifurcations, hysteresis, and bistability. The Ott–Antonsen reduction reveals that higher-order terms do not affect the linear stability of incoherence but generate explosive transitions and extensive multistability.

Figure 3: Sufficiently strong higher-order interactions induce hysteresis and bistability; the choice of nonpairwise coupling function (symmetric vs. asymmetric) yields distinct bifurcation structures and stable states.

Analytical tractability varies with the symmetry of the coupling function; symmetric triadic couplings require partial dimensionality reduction and self-consistency analysis, revealing cluster states and memory effects.

General Network Dynamics with Higher-Order Interactions

Master Stability Function Extensions

The MSF formalism, foundational for pairwise networks, has been extended to hypergraphs via generalized Laplacians. However, the presence of multiple noncommuting Laplacians complicates eigendecomposition and precludes a straightforward notion of "optimal hypergraphs" for synchronization. Dimensionality reduction remains a significant challenge, especially for systems with complex, adaptive, or multilayer higher-order structures.

Cluster Synchronization and Invariant Subspaces

Structural symmetries and graph partitions yield invariant subspaces corresponding to cluster synchrony patterns. Computational algorithms leveraging incidence matrices and block diagonalization techniques facilitate the identification and stability analysis of these patterns in higher-order networks.

Phase Reduction and Nonpairwise Interactions

Phase reduction theory connects high-dimensional oscillator dynamics to phase models with nonpairwise interactions. Higher-order terms arise from both the physical coupling structure and higher-order expansions, capturing indirect interactions and resonant effects. The persistence of invariant tori under increasing coupling strength is guaranteed up to a critical threshold, beyond which new dynamical phenomena (e.g., chaos, heteroclinic cycles) may emerge.

Figure 4: Phase reduction maps high-dimensional oscillator dynamics onto invariant tori, enabling tractable analysis of collective behaviors and the emergence of nonpairwise interactions.

Reduction and Reconstruction of Higher-Order Networks

Model Order Selection and Renormalization

The combinatorial complexity of higher-order models necessitates principled reduction strategies. The minimal order required to reproduce observed dynamics can be estimated by analyzing the functional form of interaction terms and their decomposability. Model selection frameworks balance accuracy and complexity, identifying optimal truncation orders for hypergraphs.

Renormalization approaches, inspired by RG theory, merge nodes into super-nodes based on diffusion processes and scale-invariance metrics, revealing emergent higher-order interactions even in originally pairwise networks.

Figure 5: Complexity reduction via truncation of interaction order or coarse-graining of nodes; reconstruction methods infer coupling structure from observed dynamics.

Hypergraph Inference from Time Series

Reconstruction of higher-order network structure from dynamical data is an active area, with methods ranging from linear regression (when dynamics are known) to sparse regression and phase reduction (when dynamics are unknown). Scalability remains a bottleneck, especially for interactions beyond third order and large system sizes. Derivative-free and noise-robust techniques are promising directions for future research.

Application to empirical data (e.g., EEG) demonstrates that nonpairwise interactions contribute significantly to macroscopic brain dynamics, challenging the adequacy of pairwise network models in neuroscience.

Beyond Node Dynamics: Edge and Hyperedge Dynamics

The extension of dynamical variables to edges and hyperedges, as in the simplicial Kuramoto model, leverages discrete geometry and topology. The model incorporates boundary and coboundary operators, yielding interactions from both lower and higher-dimensional faces. The discrete Hodge Laplacian governs the linearized dynamics, with harmonic modes localized along topological holes dictating synchronization properties.

Figure 6: Simplicial Kuramoto models assign phases to $k$-simplices; dynamics depend on projection to lower/higher-order simplices and the order of interaction.

Topological conditions, rather than mere connectivity, determine the existence and stability of synchronized states. The Dirac operator framework enables cross-dimensional coupling and the emergence of discontinuous transitions and spontaneous rhythms.

Adaptive higher-order interactions further enrich the dynamical landscape, influencing synchronization, opinion formation, and multiplayer game dynamics.

Implications and Future Directions

The incorporation of higher-order interactions fundamentally alters the collective dynamics of networked systems, enabling phenomena such as explosive transitions, multistability, and complex synchronization patterns. Analytical tools (e.g., multiorder Laplacians, phase reduction, MSF extensions) provide partial tractability, but scalability and generalizability remain open challenges.

Strong numerical results demonstrate that higher-order interactions can both promote and impede synchronization, depending on network representation and coupling function. The contradictory effects in hypergraphs versus simplicial complexes underscore the need for careful model selection and structural analysis.

Future research should address:

• Analytical characterization of dynamics with interactions beyond three-body terms.
• Systematic exploration of nonpairwise coupling functions and their impact on collective phenomena.
• Development of scalable, noise-robust inference algorithms for large, high-order systems.
• Identification of equivalences and reductions between different combinatorial models.
• Application of higher-order models to empirical data in neuroscience, ecology, and beyond.

Conclusion

The paper of collective dynamics on higher-order networks reveals a rich spectrum of phenomena inaccessible to pairwise models. Theoretical advances in model formulation, stability analysis, and inference are beginning to bridge the gap between abstract mathematical frameworks and real-world complex systems. As empirical data becomes increasingly available, higher-order models are poised to yield deeper insights into the organization and function of biological, social, and technological networks.