Higher-Order Network Models

• Higher-order network models are frameworks that extend traditional graphs by incorporating multiway relations, path dependencies, and group interactions.
• They are implemented using structures like hypergraphs, simplicial complexes, memory networks, and motif-based methods to accurately represent non-dyadic interactions.
• Applications in transportation, neuroscience, social systems, and other fields demonstrate that these models often reveal insights that pairwise representations overlook.

Higher-order network models generalize graphs to capture non-pairwise relations and dependencies that arise in complex systems. In the literature, the term covers combinatorial representations of multiway relations in hypergraphs and simplicial complexes, algebraic operators that extract topology and geometry, and dynamic or statistical models in which path memory, motifs, or multilayer coupling generate dependencies beyond dyadic edges (Bick et al., 2021). The concept therefore includes native polyadic structures, memory-aware state-space expansions for sequential data, and motif-based constructions in which higher-order organization is defined by recurring subgraphs rather than by hyperedges alone (Xu et al., 2015).

1. Conceptual scope and formal distinctions

A graph represents pairwise relations between vertices. Higher-order network models are introduced when that representation is insufficient because interactions involve sets of size at least three, because temporal or sequential data violate the first-order Markov assumption, or because system behavior depends on supra-dyadic organization such as motifs, cycles, and closures (Bick et al., 2021). In this sense, higher-order models extend network analysis in at least three directions: multiway relation structure, path dependence, and higher-order topology or geometry.

This conceptual breadth matters because different constructions answer different questions. Hypergraphs encode set membership relations directly, simplicial complexes add closure under faces and support topological operators, and higher-order Markov or memory networks encode ordered histories in enlarged state spaces (Schaub et al., 2021). Motif-based models occupy another position in this landscape: they treat small subgraphs as the primitive units of organization, so that adjacency, conductance, or vulnerability is defined through joint participation in recurring patterns rather than through direct edges alone (Benson, 2018).

A recurrent distinction in the literature is the difference between higher-order and multilayer modeling. Multilayer or multiplex networks represent heterogeneous channels of typically dyadic interaction, whereas higher-order models represent non-dyadic structure or memory-dependent dynamics (Bick et al., 2021). Some empirical studies combine both ideas. In multimodal transport, for example, higher-order effects can arise from motif structure and multilayer coupling without introducing hypergraphs or simplicial complexes explicitly (Song et al., 17 Sep 2025).

2. Polyadic representations: hypergraphs, simplicial complexes, and cell complexes

The most direct higher-order representation is the hypergraph. A hypergraph $H=(V,E,\omega)$ consists of a vertex set $V$, a set of hyperedges $E$, each hyperedge being a subset of vertices of cardinality at least $2$, and hyperedge weights $\omega(e)>0$ (Schaub et al., 2021). Hypergraphs may be uniform or non-uniform, directed or undirected, and heterogeneous. They preserve group interactions without forcing them into collections of pairwise edges.

Simplicial complexes impose a stronger combinatorial condition. A $k$-simplex is a set of $k+1$ vertices, and a simplicial complex is closed under taking faces: whenever a simplex is present, all of its subsets are also present (Schaub et al., 2021). This closure property is appropriate when lower-dimensional relations are implied by higher-dimensional ones. Simplicial complexes support oriented $k$-simplices, $k$-cochains, boundary operators, homology, and Hodge theory, so they serve simultaneously as data structures and as algebraic-topological objects (Bick et al., 2021).

Cell-complex models generalize simplicial complexes further by allowing higher-dimensional cells other than simplices. In Network Geometry with Flavor, higher-order networks are generated by gluing identical regular polytopes along $(d-1)$-faces, producing pure cell complexes with generalized degrees, emergent hyperbolic geometry, and polytope-dependent degree distributions (Mulder et al., 2017). This construction separates the effects of the gluing rule from those of the building block geometry, and it shows that higher-order structure can be native even when the basic cells are not simplices.

A central methodological warning is that clique or motif expansion can be lossy. Projecting an $V$0-way relation into pairwise edges discards order information, confounds within-edge dependencies, and biases centrality and mesoscale structure (Hood et al., 27 May 2025). That observation motivates native hypergraph or simplicial representations whenever the semantics of the data are fundamentally polyadic.

3. Memory networks and path-dependent higher-order models

A different family of higher-order network models addresses sequential data. In this setting, the issue is not polyadic co-occurrence but non-Markovian path dependence. The Higher-Order Network representation encodes variable orders of dependency directly into a weighted directed graph by relabeling nodes with context, so that a node such as $V$1 represents being at $V$2 given the ordered history $V$3 (Xu et al., 2015). Standard algorithms such as random walks, PageRank, centrality, and clustering then operate without modification on the expanded network.

The defining feature of this representation is variable order. The discovery procedure grows source contexts only when the next-step distribution changes sufficiently, using a Kullback–Leibler divergence criterion and support thresholds, and it rewires edges so that higher-order states receive context-consistent in-edges and transitions (Xu et al., 2015). In the shipping and clickstream datasets studied in that work, dependencies extend up to fifth and third order, respectively, whereas retweet data are found to be Markovian and therefore first-order only (Xu et al., 2015).

Multi-order generative models provide a related but distinct formulation. MOGen represents a path

$V$4

and combines dependencies up to a maximum memory $V$5 in an absorbing Markov chain with transient block $V$6, absorbing block $V$7, and start distribution $V$8 (Gote et al., 2021). Its fundamental matrix

$V$9

yields closed-form higher-order centralities, including betweenness, closeness, path end probability, continuation probability, and path reach (Gote et al., 2021). In prediction experiments on five empirical path datasets, MOGen consistently outperforms both first-order network models and direct path-based predictors; when observations per unique path are sufficiently high, the gap to the path model disappears, supporting the interpretation that network models underfit and full path models overfit (Gote et al., 2021).

Scalability is addressed explicitly in BuildHON+, which removes the need to specify MaxOrder and MinSupport. It grows contexts lazily, uses a dynamic threshold

$E$0

with default $E$1, and stops early when the maximum possible divergence cannot exceed the threshold (Saebi et al., 2017). This design yields a parameter-free higher-order representation that improves anomaly detection over first-order networks and scales to dependencies beyond fifth order in large ship-movement data (Saebi et al., 2017).

4. Motifs, multilayer coupling, and temporal higher-order interactions

Not all higher-order modeling is based on hyperedges or memory states. Motif-based approaches elevate small subgraphs to analytical primitives. Given a motif $E$2, a motif adjacency matrix $E$3 counts how often pairs of nodes jointly participate in instances of that motif, and motif conductance replaces edge conductance by counting motif instances cut by a partition (Benson, 2018). For motifs with two or three anchor nodes, the resulting objective obeys Cheeger-type bounds, and spectral clustering on the motif-induced graph yields modules that can differ sharply from edge-based partitions (Benson, 2018).

The same line of work generalizes clustering coefficients from triangle closure to higher-order closure of $E$4-wedges into $E$5-cliques, and it defines temporal motifs as ordered sequences of time-stamped edges within a time window $E$6 (Benson, 2018). This framework makes higher-order organization measurable in settings where the relevant structure is not merely group membership but a recurring relational pattern with direction, order, or sign.

Urban transport resilience provides an empirical example in which “higher-order” means organizational principles beyond dyadic edges, specifically structural motifs and multilayer coupling (Song et al., 17 Sep 2025). In a directed, multilayer multimodal transport network for Wuhan, Feed-Forward Loop motifs are used to quantify node participation in supra-dyadic patterns, while intermodal edges encode walking transfers between metro, bus, ferry, and rail layers (Song et al., 17 Sep 2025). The central result is a duality: integration improves static robustness and interoperability, yet simultaneously creates structural conduits for catastrophic cascades (Song et al., 17 Sep 2025). The same study reports that static low-order and higher-order structural metrics are poor predictors of dynamic functional resilience, with correlations to Functional Recoverability near zero for betweenness centrality ($E$7) and motif score ($E$8) (Song et al., 17 Sep 2025). This directly challenges the intuition that more elaborate static higher-order descriptors automatically capture dynamic fragility.

Temporal hypergraph prediction extends higher-order modeling to event forecasting. In a higher-order temporal network $E$9, a hyperlink $2$0 has binary activity $2$1, and prediction can be based on exponentially decayed memories of the target hyperlink and its sub- or super-hyperlinks (Jung-Muller et al., 2023). A generalized memory score

$2$2

supports one-step-ahead ranking, and across eight real-world face-to-face datasets this higher-order formulation consistently outperforms a pairwise temporal baseline, especially for orders $2$3 and $2$4 (Jung-Muller et al., 2023). A later formulation groups overlapping hyperlinks by overlap type $2$5 and learns coefficients by Lasso, improving interpretability and typically improving predictive accuracy relative to both the pairwise baseline and the more general overlap model (Peters et al., 2024).

5. Operators, spectra, and higher-order dynamics

Higher-order network analysis relies on operators that generalize graph incidence and Laplacian structure. In simplicial complexes, the boundary maps $2$6 encode incidence between $2$7- and $2$8-simplices, and the combinatorial Hodge Laplacian is

$2$9

Its down-Laplacian $\omega(e)>0$0 penalizes gradient-like variation, its up-Laplacian $\omega(e)>0$1 penalizes curl-like variation, and the corresponding Hodge decomposition splits $\omega(e)>0$2-cochains into exact, coexact, and harmonic components (Schaub et al., 2021). For $\omega(e)>0$3, this decomposition separates edge flows into gradient, curl, and harmonic subspaces, linking network analysis to topology through Betti numbers and cohomology (Schaub et al., 2021).

Hypergraphs support matrix and tensor operators as well. The Zhou normalized hypergraph Laplacian

$\omega(e)>0$4

acts on node signals in an incidence-based representation, while tensor constructions for uniform hypergraphs preserve genuinely polyadic couplings at higher computational cost (Schaub et al., 2021). The trade-off between matrix and tensor formulations is a recurring theme: matrix methods are scalable and compatible with graph toolchains, whereas tensor methods are more expressive but computationally harder and less mature (Schaub et al., 2021).

These operators support dynamics. On simplicial complexes, denoising, interpolation, filtering, embedding, and neural architectures can be defined directly on $\omega(e)>0$5-cochains using $\omega(e)>0$6 (Schaub et al., 2021). In temporal networks, higher-order models also affect controllability. By representing time-respecting paths with a second-order network and examining spectral properties of the resulting higher-order Laplacian, it becomes possible to explain why non-Markovian ordering can either increase or decrease the minimum time needed to control a temporal system (Zhang et al., 2017). The relevant observation is qualitative rather than a closed-form law: empirical order correlations that decrease the higher-order algebraic connectivity tend to slow controllability, whereas correlations that increase it tend to speed it up (Zhang et al., 2017).

Percolation on temporal hypergraphs provides another dynamic example. In the higher-order activity-driven model, the integrated connection probability for an $\omega(e)>0$7-order hyperlink up to time $\omega(e)>0$8 is

$\omega(e)>0$9

which leads to analytical expressions for expected hyperdegree, degree correlations, and correlated or uncorrelated percolation times (Gaetano et al., 2023). The same framework shows that projecting higher-order interactions onto pairwise networks underestimates percolation time, with the discrepancy growing with interaction order (Gaetano et al., 2023).

6. Statistical inference, mesoscale structure, and higher-order learning

A major development in higher-order network modeling is the shift from descriptive constructions to identifiable statistical models. In generalized latent space models, a triadic interaction tensor is parameterized by latent positions and a multilinear core through

$k$0

with $k$1 (Lyu et al., 2021). This framework subsumes hypergraph latent space models, mixture multilayer latent space models, and dynamic latent space models, and it is estimated by generalized tensor decomposition with projected gradient descent on Grassmannians (Lyu et al., 2021). The analysis establishes linear convergence under mild conditions and finite-sample error rates determined by signal strength, degrees of freedom, and link smoothness (Lyu et al., 2021).

Probabilistic hypergraph models have also moved beyond strictly assortative block structure. A recent class of models treats classes of similar units as nodes in a latent hypergraph and factorizes class-level affinity tensors through a low-rank symmetric CP representation, yielding a broad spectrum from assortative to disassortative mesoscale structure while preserving identifiability (Hood et al., 27 May 2025). The key rate expression is

$k$2

with $k$3 combining node-class and class-community memberships (Hood et al., 27 May 2025). Empirically, this model improves link prediction over a strictly assortative baseline and yields lower-entropy, hence sharper, node memberships in most datasets examined (Hood et al., 27 May 2025).

Order dependence itself can be modeled explicitly. HyperMOSBM relaxes the assumption that one affinity pattern governs all hyperedge sizes by partitioning the set of interaction orders into subsets, each with its own affinity matrix, and selecting the partition that maximizes out-of-sample hyperlink prediction AUC (Nakajima et al., 26 Nov 2025). Its Poisson rate

$k$4

depends on the order-specific block structure chosen for $k$5 (Nakajima et al., 26 Nov 2025). Across a diverse set of real-world hypergraphs, multi-order block structures are prevalent, and the multi-order model typically improves predictive performance over the single-order baseline while producing sharper community assignments (Nakajima et al., 26 Nov 2025).

Higher-order learning also appears in neural operators and probabilistic message passing. Higher-order convolution for retinal response prediction embeds quadratic Volterra-like terms directly inside a local spatiotemporal convolution,

$k$6

improving neural predictivity and sample efficiency without increasing depth (Azeglio et al., 12 May 2025). Neuralized higher-order belief propagation instead represents higher-order factors by low-rank CP tensors so that factor-to-variable messages reduce to matrix multiplications and Hadamard products, with total per-iteration complexity $k$7 (Dupty et al., 2020). Spectral Higher-Order Neural Networks push the same theme into general-purpose feedforward architectures, using spectral parameterization of triadic interactions to reduce parameter growth from $k$8 to $k$9 while improving stability (Peri et al., 30 Mar 2026).

7. Applications, recurrent findings, and methodological limits

Higher-order network models have been applied across transportation, social interaction, neuroscience, biology, economics, communication, and machine learning. In transportation, path-driven higher-order analysis of the classical Sioux Falls benchmark shows limited path diversity, rapid fragmentation at higher orders, and weak alignment with empirical routing behavior, whereas an extended Sioux Falls network aligns much more closely with trajectory-derived higher-order structure (Zhang et al., 8 Aug 2025). In resilient-city modeling, higher-order motifs and multilayer coupling reveal a paradox in which integration improves static robustness while worsening dynamic cascade vulnerability (Song et al., 17 Sep 2025). In temporal contact networks, hyperlink-level memory models show that self-history and overlap with sub- and super-hyperlinks materially improve one-step-ahead prediction of higher-order events (Peters et al., 2024).

Several recurrent findings cut across these domains. First, pairwise reductions routinely underfit. This is explicit in sequential modeling, where first-order networks misrepresent path statistics (Xu et al., 2015), in temporal prediction of group interactions, where pairwise baselines lose signals from nested or overlapping group structures (Jung-Muller et al., 2023), and in hypergraph modeling, where clique expansion discards order-specific information (Hood et al., 27 May 2025). Second, unrestricted high-order descriptions can overfit. MOGen was designed precisely to sit between first-order underfitting and full path-model overfitting (Gote et al., 2021). Third, “higher-order” by itself does not guarantee better prediction of dynamics: static motif participation can be less informative than random-failure baselines for some resilience tasks, and structural higher-order metrics may fail to track functional load redistribution (Song et al., 17 Sep 2025).

Methodological limits differ by family. Hypergraph and tensor models face combinatorial growth in states, factors, or possible hyperedges (Schaub et al., 2021). Memory networks depend on observed sequential data and can become difficult to interpret after state splitting (Xu et al., 2015). Temporal hypergraph predictors may assume known candidate hyperlinks or known event counts per order at prediction time (Jung-Muller et al., 2023). Urban cascade models may rely on shortest-path flow estimation, static topology, and capacities proportional to inferred stationary load rather than on adaptive behavior or schedules (Song et al., 17 Sep 2025). These limitations do not reduce the importance of higher-order models; they specify the trade-off between fidelity, interpretability, and computational cost.

A consistent implication is that model choice must match interaction semantics. Hypergraphs and simplicial complexes are appropriate when the basic object is a group or a face relation; memory networks are appropriate when order matters along paths; motif-induced graphs are appropriate when the key structure is a repeated supra-dyadic pattern; and multi-order statistical models are appropriate when different interaction sizes obey different mesoscale mechanisms (Bick et al., 2021). Across these settings, higher-order network models function less as a single formalism than as a principled class of representations and operators for systems in which dyadic structure is not enough.