Higher-Order Interactions in Complex Systems

• Higher-order interactions are multi-way dependencies among three or more entities that capture non-additive, context-dependent effects in complex systems.
• They are modeled using advanced mathematical tools like hypergraphs, simplicial complexes, and composite Laplacians to quantify structural and dynamical properties.
• HOIs drive breakthroughs in synchronization, ecological stability, and neural computation, enhancing both theoretical insights and practical applications.

Higher-order interactions (HOIs)—defined as dynamical, structural, or statistical dependencies involving three or more entities simultaneously—are fundamental to the description of collective phenomena in complex systems across the physical, biological, ecological, and informational sciences. Unlike pairwise or dyadic interactions, HOIs capture non-additive effects and context-dependent behavior that arise when the combined influence of multiple nodes, variables, or species cannot be decomposed into a sum of individual effects. Recent advances have formalized the mathematical, algorithmic, and empirical significance of HOIs through tools such as hypergraphs, simplicial complexes, multi-order Laplacians, information-theoretic metrics, and higher-order dynamical models. HOIs have been shown to fundamentally alter system-level outcomes such as synchronization, robustness, ecological stability, memory capacity, and species abundance patterns.

1. Mathematical Representation and Structural Foundations

The formalization of HOIs relies on mathematical structures that can natively encode interactions involving arbitrary numbers of entities. Hypergraphs represent HOIs as hyperedges (subsets of nodes), while simplicial complexes impose inclusion constraints so that all subgroups of an interacting set are simultaneously coupled (Zhang et al., 2022). These representations differ in the constraints they encode: hypergraphs allow arbitrary group interactions, whereas simplicial complexes guarantee that the presence of a higher-order interaction (simplex) implies the existence of all lower-order sub-interactions among its constituents.

In certain applications, more expressive models such as pangraphs (or “ubergraphs”) generalize even further by allowing interactions to themselves act as vertices in higher nodes, enabling the nesting of interactions and the explicit modeling of asymmetric or role-specific HOIs (Iskrzyński et al., 14 Feb 2025). The incidence matrix H, frequently used in hypergraph contexts, captures node-to-hyperedge relationships and serves as a foundational object for constructing algorithmic and spectral tools (Kim et al., 1 Apr 2024).

These structures facilitate the definition of degree, centrality, and clustering measures that capture the role and importance of nodes, hyperedges, or even nested interactions in higher-order network topology. Weighted and real-valued incidence matrices allow quantification of interaction strengths and roles, critical for modeling asymmetric or context-dependent multi-way effects.

2. Dynamical Systems and Collective Behavior

HOIs fundamentally reshape the collective dynamics of systems by introducing non-additive coupling terms into the governing equations. In networks of coupled oscillators, for example, triadic or higher-order couplings augment the standard Laplacian operator, leading to a composite Laplacian of the form

$L = (1 - \alpha) L^{(1)} + \alpha L^{(2)},$

where $L^{(1)}$ is the normalized Laplacian for pairwise (dyadic) interactions and $L^{(2)}$ encodes triadic (clique-based or simplex) interactions (Skardal et al., 2021).

Broadening the eigenvalue spectrum of this composite Laplacian, achieved by strengthening the higher-order term (increasing $\alpha$), optimizes the alignment between node-intrinsic dynamics and network structure. Specifically, a broadened spectrum increases the dominant eigenvalue $\lambda_N$ and can decrease the smallest nontrivial eigenvalue $\lambda_2$. The effect is a reduction in synchronization error when natural frequencies optimally align with network modes, which enhances the system’s ability to synchronize and widens the dynamical range.

The effect of HOIs is strongly representation-dependent: in random hypergraphs, HOIs tend to homogenize the degree distribution and enhance synchronization, while in simplicial complexes, the inclusion property drives increased degree heterogeneity and can destabilize synchronization (Zhang et al., 2022). This underscores the necessity of selecting appropriate structural models to faithfully capture dynamical consequences of HOIs.

3. Information-Theoretic Quantification of HOIs

Information-theoretic measures extend beyond pairwise mutual information to capture the redundancy and synergy that emerge in multivariate systems. Central to this effort is the O-information, a functional that quantifies whether collective dependencies among multiple variables (or time series) are dominated by redundancy (positive O-information) or synergy (negative O-information) (Faes et al., 2022). The recursive definition

$$\Omega_{X^N} = \Omega_{X^{N \setminus j}} + \Delta_{X^{N \setminus j}; X_j}$$

with increment

$$\Delta_{X^{N \setminus j}; X_j} = (2-N) I(X_j; X^{N \setminus j}) + \sum_{m \neq j} I(X_j; X^{N \setminus \{m, j\}})$$

generalizes mutual information rate to the multivariate case and can be decomposed into Granger-causal and instantaneous terms, with frequency-domain extensions enabling analysis in multirhythmic systems.

This framework enables a multiresolution analysis: at the global level (entire network), the O-information and its rate generalizations identify the balance of synergy and redundancy; at the link and node level, local O-information and O-information gradients dissect contributions of individual interactions or subsystems (Mijatovic et al., 28 Aug 2024).

For practical extraction of HOIs in high-dimensional biological data (e.g., fMRI), matrix-based Rényi $\alpha$-order entropy estimators are used to efficiently compute the total and dual total correlation, facilitating computation of O-information tensors that quantify HOIs among all triplets (or higher-order groups) of regions (Zhang et al., 3 Jul 2025).

4. HOIs in Ecological and Evolutionary Dynamics

Classical ecological models, such as the Lotka–Volterra (LV) and replicator dynamics frameworks, have been generalized to include explicit higher-order terms (Cui et al., 28 May 2024, Kang et al., 29 Jul 2025). The generalized GLV model

$$\frac{dN_i}{dt} = N_i \bigg( r_i + \sum_{j=1}^S \alpha_{ij} N_j + \sum_{j,k=1}^S \beta_{ijk} N_j N_k \bigg) + d_i$$

enables the modeling of indirect or context-dependent species influences. Analysis using tensor algebra (employing S-tensors and polynomial complementarity problems) yields existence, uniqueness, and stability conditions for positive equilibria.

Incorporation of HOIs profoundly alters dynamical outcomes:

• Even weak HOIs can prevent collapse under high variability in pairwise interactions, stabilizing or rescuing ecological communities (Kang et al., 29 Jul 2025).
• HOIs can drive systems into oscillatory, quasi-periodic, or chaotic regimes, supporting persistent coexistence and empirical species abundance distributions not captured by pairwise-only models.
• The dynamic onset speed of HOIs (modulated by response delay parameters such as $\omega$) can induce structural changes, oscillations, or extinctions depending on the network’s intransitivity and the interplay between timescales (Giel et al., 26 Aug 2024).

From an evolutionary perspective, higher-order interactions facilitate cooperation more readily than pairwise-only structures. In hypernetwork game models, the critical benefit-to-cost threshold for cooperation is lowered by increasing the size (order) or density (hyperdegree) of higher-order groups (Guo et al., 11 Jan 2025), with the rule

$b/c > \frac{d \cdot g}{d + 1}$

(d hyperdegree, g hyperedge order) indicating a less stringent requirement than classical pairwise games.

5. Empirical Analysis and Predictive Modeling

HOIs are empirically extracted and analyzed using hypergraphs, where a hyperedge indicates the co-appearance of a group. HOI persistence over time (the number of time steps a group reappears) is a key indicator of the strength and robustness of collective relationships (Choo et al., 2022). Persistence distributions typically obey power laws with exponents that decrease (i.e., heavier tails) as the size of the interaction group increases, meaning that larger HOIs are less likely to be persistent. Random forest regression and similar models can accurately predict HOI persistence using structural features such as the number of containing hyperedges and the average weighted degree of neighbors.

Filtering frameworks isolate scale-dependent patterns by slicing hypergraphs according to hyperedge size and analyzing each “layer” with measures of effective information, assortativity, and centrality—revealing that nodal importance, community structures, and even global connectivity patterns are stratified by interaction scale (Landry et al., 2023).

In time-dependent settings, memory-based models show that the future occurrence of HOIs (e.g., group contacts) is best predicted by a combination of recent activity of the group itself, its sub-hyperlinks, and super-hyperlinks—demonstrating the importance of both local self-memory and context provided by nested HOIs (Peters et al., 9 Aug 2024).

6. Implications for Learning, Computation, and Model Selection

HOIs have driven methodological advances in both machine learning and network science. Hypergraph neural networks (HNNs) directly exploit HOIs via dedicated message passing, structural transformations, and training objectives that recognize the multi-way nature of the input (Kim et al., 1 Apr 2024). Message-passing strategies in HNNs aggregate information along hyperedges, supporting rich representation learning for applications ranging from bioinformatics and recommendation systems to computer vision.

In neural network interpretability and learning, total correlation and O-information serve as natural objectives for uncovering local or global HOIs among hidden units, revealing which latent factors or neuronal groups encode specific features or redundancies (Kerby et al., 6 Feb 2024, Zhang et al., 3 Jul 2025). Curved neural network models, developed by introducing a deformed entropy function (e.g., Rényi or Tsallis entropy), provide a mathematically tractable route to induce effective higher-order couplings, thereby yielding tractable models with enhanced memory retrieval capacity, explosive phase transitions, and robust basins of attraction (Aguilera et al., 5 Aug 2024).

HOI representational choices critically influence analytical results, with model selection (hypergraph, simplicial complex, pangraph) directly affecting the spectral, dynamical, and centrality properties derived from the network—altering our identification of pivotal agents or groupings in real-world systems (Zhang et al., 2022, Iskrzyński et al., 14 Feb 2025). Stepwise reduction and filtering approaches offer data-adaptive model complexity tuning, essential for managing computational cost and interpretability while maximizing empirical performance in tasks such as link prediction or community detection (Bian et al., 8 Nov 2024).

7. Future Prospects and Open Challenges

The field’s primary challenges include generalizing theoretical results to heterogeneous or temporally evolving systems, rigorously detecting and parameterizing HOIs in high-dimensional data, and developing efficient computational frameworks for large systems. Empirical results show that the benefit of HOI modeling is domain- and context-specific, with some networks deriving marked improvements in predictive or descriptive accuracy, while others are well described by lower-order models (Bian et al., 8 Nov 2024).

Advancements are anticipated in several directions:

• Deeper information-theoretic theory connecting multivariate redundancy/synergy patterns with structural network features
• Dynamic and adaptive models that recognize the context-dependence and temporal activation of HOIs
• Integration of HOIs with temporal, spatial, and heterogeneous networks in both deterministic and stochastic settings
• Enhanced learning algorithms and neural architectures capable of leveraging, explaining, and visualizing complex higher-order patterns in massive datasets

By rigorously characterizing, quantifying, and modeling HOIs, researchers advance toward a more holistic, accurate, and robust understanding of complex systems that surpasses the limitations of pairwise-only frameworks.