Higher-Order Networks

Updated 2 July 2026

Higher-Order Networks are mathematical models that capture interactions among groups rather than just pairs, enabling the representation of complex, non-Markovian dependencies.
They encompass various frameworks such as hypergraphs, simplicial complexes, and variable-order Markov models to model diverse forms of higher-order interactions.
Advanced algorithms and spectral methods applied to HONs facilitate robust community detection, flow prediction, and the analysis of dynamical phenomena in complex systems.

A higher-order network (HON) is a class of mathematical models that generalize conventional graphs by encoding interactions among arbitrary-size groups of entities or by faithfully representing non-Markovian, variable-order dependencies in sequential data. Higher-order networks have become a central paradigm for modeling complex systems whose interactions cannot be reduced to dyads, encompassing fields such as network science, machine learning, neuroscience, and dynamical systems. Their structural and algorithmic diversity spans hypergraphs, simplicial complexes, motif-based models, and variable-order Markovian constructions, each capturing fundamentally different forms of higher-order structure (Bick et al., 2021, Tian et al., 2024). This entry reviews the theoretical underpinnings, key modeling frameworks, analysis methods, dynamical phenomena, and leading application domains for higher-order networks.

1. Formal Definitions and Structural Paradigms

A higher-order network is any combinatorial object in which the primitive interactions can involve more than two entities. The field has adopted two principal representations:

Hypergraphs: An undirected hypergraph is a pair $H=(V, E)$ , where $V$ is a set of $N$ nodes and $E\subseteq 2^V\setminus\{\emptyset\}$ is a collection of hyperedges, each a subset of $V$ of size $\geq 2$ . Hyperedges capture group interactions, e.g., co-authorship teams in collaboration networks (Majhi et al., 2022).
Simplicial Complexes: A simplicial complex $K$ on $V$ is a set of nonempty subsets ("simplices") closed under downward inclusion: if $\sigma\in K$ , every face $\tau\subset\sigma$ also belongs to $V$ 0. This hierarchical model ensures that all subgroups of an interaction are included as lower-dimensional interactions, thereby capturing inclusion structure (Landry et al., 2023).

Alternative paradigms include motif-based models, where the focus is on small frequently occurring subgraphs (e.g., triangles, cliques) in ordinary networks, and variable-order networks built from sequential data, where nodes encode variable-length contexts to represent non-Markovian dependencies (Tian et al., 2024, Xu et al., 2015). In such sequential HONs, a node might encode the fact that a system is in state $V$ 1 having arrived via a particular path, enabling precise modeling of higher-order transition rules (Saebi et al., 2017, Chen et al., 2023).

The order of an interaction refers to its cardinality; traditional graphs encode only order-2 (pairwise) edges. HONs generalize to arbitrary order $V$ 2.

2. Algorithmic Construction and Model Inference

The construction of HONs depends on the data modality:

From Sequential Data: Algorithms such as BuildHON and BuildHON+ discover variable-order dependencies from trajectory data. Nodes correspond to distinct $V$ 3-gram contexts for all $V$ 4 where higher-order statistics are significant, determined by Kullback–Leibler divergence tests between successive context distributions. Edges encode empirical transition probabilities. HONs thus emerge as directed graphs in which context nodes are created precisely when higher-order memory is statistically supported, leading to compact, faithful representations that retain compatibility with standard network analysis algorithms (Xu et al., 2015, Saebi et al., 2017, Chen et al., 2023, Saebi et al., 2019).
From Relational Data: When interactions or events already occur among groups (e.g., co-authorship lists, social gatherings), hyperedges are constructed by collecting the relevant sets. Simplicial complexes are obtained by closing the observed hyperedges under downward inclusion.
Motif-Based Extraction: For dyadic graphs, higher-order structure can be revealed by explicit motif enumeration, constructing motif adjacency matrices or motif Laplacians for further spectral analysis and clustering (Tian et al., 2024).
Stochastic Block Models (SBMs) for Hypergraphs: Mesoscale community structure in observed hypergraphs often depends on the order of interaction. Multi-order block models infer partitions of node sets and simultaneously fit order-dependent affinity matrices, with optimal grouping determined via cross-validated link prediction AUC (Nakajima et al., 26 Nov 2025).

Empirical N-gram Collection: Count all observed $V$ 5-grams up to some maximum order; prune using minimum support.
Significance Testing: For each extension $V$ 6 of a context $V$ 7, compute $V$ 8; retain higher-order nodes if divergence exceeds threshold.
Network Wiring: Nodes are variable-length contexts; edges represent statistically valid transitions. Edge rewiring ensures transitions always point to the most-specific representation.

3. Structural and Topological Features

HONs exhibit key structural properties not present in dyadic graphs:

Inclusion and Simpliciality: In simplicial complexes, all subfaces of an interaction must exist as lower-order interactions (downward closure). Most real data, however, are neither fully simplicial nor fully arbitrary; systems typically show partial simpliciality quantified by measures such as:
- Simplicial Fraction (SF): Proportion of hyperedges that are downward closed.
- Edit Simpliciality (ES) and Face Edit Simpliciality (FES): Fraction of necessary subfaces present, globally and per maximal face (Landry et al., 2023).
Group-Size Distribution and Overlaps: In hypergraphs, the distribution of group sizes and the extent of node co-membership (overlap) strongly influence cohesion, fragmentation, and modularity (Filho, 2022).
Generalized Degree Distributions: Node and simplex degrees in HONs often follow scale-free or "bending power law" distributions. In NGF and BPL models, the degree exponent is explicitly controlled by the model's attachment kernel and simplex dimension (Guo et al., 2023, Mulder et al., 2017).

4. Analysis Methods and Learning Algorithms

Analysis of HONs leverages extended versions of classical tools:

Incidence and Adjacency Matrices/Tensors: Hypergraphs and simplicial complexes are encoded via incidence matrices and higher-order adjacency tensors, permitting generalization of Laplacian and spectral methods (Majhi et al., 2022).
Hodge Laplacian and Persistent Homology: On simplicial complexes, boundary operators and their Laplacians offer access to homological features, including Betti numbers and higher-dimensional holes. Persistent homology tracks the evolution of these features across scales (Mulder et al., 2017, Tian et al., 2024).
Motif Spectral Methods: Motif Laplacians and motif conductance-based clustering identify functional modules that coincide with higher-order interaction patterns (Tian et al., 2024).
Higher-Order Embedding: HONEM and related approaches construct low-dimensional embeddings for HONs by assembling higher-order neighborhood matrices encompassing all statistically valid transition probabilities, followed by truncated SVD or tensor decomposition (Saebi et al., 2019, Tian et al., 2024).
Random Walks, PageRank, and Markov Processes: Standard dynamical measures (random walks, PageRank) can be directly applied to HONs. Care must be taken in sequential HONs to normalize for variable node representation or adapt teleportation to avoid bias (Coquidé et al., 2021).

5. Dynamical Processes and Emergent Phenomena

Dynamical systems on HONs yield behaviors fundamentally distinct from those on classical graphs:

Synchronization: Higher-order coupling (e.g., triadic terms in the Kuramoto model) can induce abrupt, multistable, or explosive synchronization transitions not captured by pairwise models (Majhi et al., 2022).
Contagion: Epidemic thresholds and outbreak regimes can change qualitatively, with possible bistability and discontinuous transitions when group infection terms are nonlinear (Majhi et al., 2022).
Evolutionary Games: Genuine multiplayer interactions modify the regions of cooperative stability in public goods and social dilemma scenarios (Majhi et al., 2022).
Consensus Formation: The convergence rate and consensus value can be strongly affected by higher-order interactions, sometimes leading to outcomes not interpolatable from pairwise consensus (Majhi et al., 2022, Krishnagopal et al., 2023).

Computational and analytic models of HONs have established that hypergraph and simplicial topologies shape all aspects of dynamical regimes, including the speed and stability of diffusion processes and the capacity for topological signal propagation (Mulder et al., 2017, Krishnagopal et al., 2023).

6. Applications and Empirical Findings

HONs are widely used across disciplines:

Transportation and Flow Systems: HONs model non-Markovian dependencies in maritime traffic and particle flows in turbulent systems. HON-based representations outperform first-order approaches in anomaly detection, partitioning, and flow prediction (Saebi et al., 2017, Chen et al., 2023).
Neuroscience and Biology: Simplicial complexes capture functional motifs and structural holes in neural connectomes, microbial communities, and protein interaction networks (Majhi et al., 2022, Tian et al., 2024, Krishnagopal et al., 2023).
Social and Information Networks: Multi-way group structure and motif organization underlie community detection, influence propagation, and recommendation systems (Tian et al., 2024).
Machine Learning and Neural Architectures: Higher-Order Neural Networks (e.g., Spectral Higher-Order Neural Networks, Higher-Order Attention in Transformers) natively encode triadic or multi-token correlations, breaking expressivity barriers with little parameter overhead (Chen et al., 3 Dec 2025, Peri et al., 30 Mar 2026).
Mesoscale Organization: Multi-order block models reveal that real hypergraphs often have order-dependent community structure, not captured in dyadic or single-order models (Nakajima et al., 26 Nov 2025).

Empirical studies show that real systems span the full range between arbitrary hypergraphs and fully simplicial complexes; most lie at intermediate values of simpliciality (Landry et al., 2023).

7. Open Challenges and Research Directions

Unresolved questions drive ongoing research:

Generative Modeling: Existing random hypergraph and block models fail to reproduce observed inclusion and overlap patterns. New models must incorporate explicit controls for inclusion/simpliciality and overlap parameters (Landry et al., 2023, Filho, 2022).
Optimal Order Discovery: Adaptive strategies for selecting context order in variable-order sequential HONs and for partitioning interaction orders in block models improve both predictive performance and interpretability (Saebi et al., 2017, Nakajima et al., 26 Nov 2025).
Scalability: Computational challenges in enumeration, embedding, homology, and SBMs require algorithmic advances, particularly for large-scale systems with rich higher-order structure (Tian et al., 2024).
Dynamics on Multiplex and Temporal HONs: Layer-coupled or time-evolving HONs present unsolved questions about signal diffusion, controllability, and interplay between topological layers (Krishnagopal et al., 2023, Majhi et al., 2022).
Learning and Inference: Efficient recovery of HON structure from observational data—especially in the presence of noise or partial observability—remains a critical inverse problem (Majhi et al., 2022).

Adoption of higher-order networks continues to increase across fields as their unique descriptive and predictive power over dyadic models is empirically validated. Accurate modeling of real-world polyadic systems now routinely requires higher-order representations and analysis.

Key references: (Bick et al., 2021, Tian et al., 2024, Majhi et al., 2022, Xu et al., 2015, Saebi et al., 2017, Chen et al., 2023, Landry et al., 2023, Mulder et al., 2017, Peri et al., 30 Mar 2026, Chen et al., 3 Dec 2025, Nakajima et al., 26 Nov 2025, Guo et al., 2023, Filho, 2022, Coquidé et al., 2021, Saebi et al., 2019, Krishnagopal et al., 2023).