Networks beyond pairwise interactions: structure and dynamics

Federico Battiston; Giulia Cencetti; Iacopo Iacopini; Vito Latora; Maxime Lucas; Alice Patania; Jean-Gabriel Young; Giovanni Petri

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Networks beyond pairwise interactions: structure and dynamics

Federico Battiston, Giulia Cencetti, Iacopo Iacopini, Vito Latora, Maxime Lucas, Alice Patania, Jean-Gabriel Young, Giovanni Petri

arXiv:2006.01764v1Jun 2, 2020physics.soc-phcond-mat.dis-nncs.SInlin.AOq-bio.NC

TL;DR

Many complex systems involve group interactions that pairwise networks cannot represent adequately. This review unifies higher-order representations, measures, models, and dynamics, showing that higher-order structure can produce distinct structural and dynamical behavior.

Problem
Pairwise network models can oversimplify systems whose biological, social, or ecological interactions involve groups of three or more nodes.
Method
The paper reviews frameworks for explicitly representing, measuring, modeling, and analyzing higher-order systems and their dynamics.
Results
Higher-order models differ substantially from pairwise models structurally and dynamically, with higher-order interactions changing critical behavior and enabling improved interaction prediction and signal denoising.
Takeaways & Limitations
Higher-order topology introduces new nonlinearities and state variables while linking structural organization to dynamical behavior across complex systems.
Takeaways & Limitations
Temporal, multiplex, and multilayer measures remain underdeveloped, leaving substantial space for future work.

Abstract

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The complexity of many biological, social and technological systems stems from the richness of the interactions among their units. Over the past decades, a great variety of complex systems has been successfully described as networks whose interacting pairs of nodes are connected by links. Yet, in face-to-face human communication, chemical reactions and ecological systems, interactions can occur in groups of three or more nodes and cannot be simply described just in terms of simple dyads. Until recently, little attention has been devoted to the higher-order architecture of real complex systems. However, a mounting body of evidence is showing that taking the higher-order structure of these systems into account can greatly enhance our modeling capacities and help us to understand and predict their emerging dynamical behaviors. Here, we present a complete overview of the emerging field of networks beyond pairwise interactions. We first discuss the methods to represent higher-order interactions and give a unified presentation of the different frameworks used to describe higher-order systems, highlighting the links between the existing concepts and representations. We review the measures designed to characterize the structure of these systems and the models proposed in the literature to generate synthetic structures, such as random and growing simplicial complexes, bipartite graphs and hypergraphs. We introduce and discuss the rapidly growing research on higher-order dynamical systems and on dynamical topology. We focus on novel emergent phenomena characterizing landmark dynamical processes, such as diffusion, spreading, synchronization and games, when extended beyond pairwise interactions. We elucidate the relations between higher-order topology and dynamical properties, and conclude with a summary of empirical applications, providing an outlook on current modeling and conceptual frontiers.

I. INTRODUCTION … III. MEASURES

The paper argues that pairwise networks cannot represent genuine group interactions, motivating explicit higher-order frameworks such as simplicial complexes and hypergraphs. It reviews their representations, relationships, limitations, and structural measures for understanding complex systems.

I. INTRODUCTION: Complex-system behavior depends on nonlinear interactions among components, and pairwise networks miss group interactions central to social, ecological, and biological systems.Higher-order interactions are therefore needed to explain and predict collective behaviors that dyadic descriptions cannot capture.
1. Low- versus high-order representations: Higher-order interactions involve groups of more than two nodes, whereas low-order systems contain only self- or pairwise interactions.The review excludes multilayer link types and non-Markovian temporal dependencies from this higher-order definition.
2. Graph-based representations: Graph projections replace each group interaction with pairwise edges, enabling standard network analysis but making the original group structure unrecoverable.Consequently, inferred higher-order structures from cliques or communities can be incomplete or misleading.
PAIRWISE REPRESENTATION: Bipartite graphs preserve membership between nodes and interactions, while projections turn each interaction into a clique and lose group structure.Motifs add recurring interaction patterns, but their number grows exponentially; cliques can also create higher-order interactions absent from the original data.
3. Explicit higher-order representations: Simplicial complexes encode higher-order interactions explicitly while requiring every simplex’s subfaces, supporting dimension-specific Laplacians, boundary operators, and topological-hole analysis.This subface constraint is useful for some group interactions but too restrictive when groups exist without functional pairwise subsets.
3. Explicit higher-order representations: Hypergraphs provide the most general representation because hyperedges need not include their subinteractions, although this flexibility complicates operators and applications.They can even contain hyperedges within hyperedges.
B. Relations and links between representations: Hasse diagrams expose simplex-inclusion hierarchies and support walks, while facet representations compactly encode complexes and correspond to constrained bipartite or hypergraph descriptions.Operators associated with different representations are not necessarily equivalent, so representation choice affects structural and dynamical analysis.
III. MEASURES: Section III introduces matrices, tensors, and generalized graph measures for quantifying structural properties of higher-order systems across cliques, hyperedges, sets, and simplices.The measures are intended to characterize each representation level and extract structural insights.

A. Matrix representations of higher-order systems … 2. Paths and path-based centralities

Higher-order systems can be represented through incidence matrices, adjacency matrices, tensors, and dimension-specific simplicial adjacencies that preserve their group structure. These representations support degree, walk, path, and centrality measures whose connectivity depends on intersection rules, simplex dimension, and whether the system is a hypergraph or simplicial complex.

1. Incidence matrix: Incidence matrices encode node–interaction membership in graphs, hypergraphs, and simplicial complexes, with weighted entries accommodating repeated vertices in hyperedges.They are also equivalent to adjacency matrices of bipartite graphs linking nodes to interactions.
1. Incidence matrix: Node degrees and hyperedge or simplex sizes follow directly from row and column sums of the incidence matrix.In graphs, each edge column sums to two, whereas higher-order interactions can involve more than two vertices.
2. Adjacency matrix: Adjacency representations count shared hyperedges between nodes, shared vertices between interactions, or motif-specific co-memberships, and can incorporate interaction weights.Dimension-specific adjacency matrices and tensors isolate higher-order structure, while intersection profiles quantify overlap between hyperedges [85].
2. Adjacency matrix: Simplicial complexes require adjacency definitions that preserve relations across dimensions, including lower adjacency, upper adjacency, and lower adjacency excluding upper adjacency.These constructions avoid listing every subsimplex while retaining the relationships needed for simplicial walks and centralities [93].
1. Degree centralities: Degree centrality generalizes through adjacency matrices and tensors to capture node relationships across higher-order dimensions, while hypergraph weights may represent incident interactions or adjacent-node ties.Hyperedge or simplex centralities can likewise use lower, upper, or combined adjacency degrees [93] [99].
2. Paths and path-based centralities: Path-based centralities extend graph definitions of betweenness, closeness, and subgraph centrality by defining walks between hyperedges or simplices rather than directly between vertices.A basic hypergraph walk uses successive hyperedges sharing at least one vertex, while subhypergraph centrality counts closed walks [82, 101].
2. Paths and path-based centralities: k-walks require successive hyperedges to intersect in at least k vertices, with all participating hyperedges having dimension at least k + 1 but no fixed maximum dimension [91, 102].In simplicial complexes, k-walks are more specifically restricted to lower-adjacent k-simplices, yielding simplicial closeness and harmonic closeness measures [99].
2. Paths and path-based centralities: Restricting intersection size or simplex dimension changes higher-order connectivity: one-vertex walks recover underlying-graph connectedness, whereas stricter overlaps can disconnect interactions and distinguish hypergraphs from simplicial complexes.For example, requiring at least two shared vertices separates the triangle from the tetrahedron [3] [4], while simplicial subfaces create additional walk constraints.

3. Eigenvector centralities … 2. Evolving simplicial complexes

The paper extends centrality, clustering, and topological analysis beyond pairwise networks to hypergraphs and simplicial complexes. It develops algebraic and persistent-homology tools for characterizing higher-order structure, including evolving complexes and dimensional holes.

3. Eigenvector centralities: Eigenvector centrality generalizes from node connectivity to bipartite graphs, hypergraphs, motifs, and tensors, enabling scores for both vertices and higher-order interactions.Incidence matrices yield centrality scores for vertices and hyperedges, while tensor eigenvectors model multiplicative contributions within hyperedges.
C. Triadic closure and clustering coefficient: Higher-order clustering coefficients preserve triadic-closure ideas by counting closed paths, cliques, simplices, or motifs under representations suited to hypergraphs and simplicial complexes.In bipartite graphs, clustering can instead use one-mode projections, where 4-paths participating in 6-cycles provide the relevant closure structure.
D. Simplicial homology: Simplicial homology studies higher-order structure across dimensions after orienting simplices, providing an algebraic framework for analyzing simplicial complexes.Orientations are arbitrary vertex-ordering choices needed for coherent computations and are irrelevant for 0-simplices.
1. Boundary operators and homology groups: Chains and boundary operators connect k-simplices to their faces, with kernels defining cycles and images defining boundaries through the relation ∂k ○∂k+1 = 0.For a triangle, the boundary is the alternating sum of its three oriented edges.
1. Boundary operators and homology groups: Homology groups identify cycles that are not induced by boundaries as higher-dimensional holes, while Betti numbers classify connected components, cycles, and voids by dimension.The 0th Betti number counts connected components, the 1st counts cycles, and higher Betti numbers represent higher-dimensional voids.
2. Evolving simplicial complexes: Homology has been extended to weighted and growing simplicial complexes, supporting topological analysis of evolving higher-order systems.These extensions build on the established algebraic-topological framework for classifying shapes.
2. Evolving simplicial complexes: Persistent homology computes homology across filtrations of growing complexes, interpreting features that persist across scales as potentially more relevant to the data space.That interpretation depends crucially on how the filtration is constructed.
2. Evolving simplicial complexes: Zig-zag and multipersistent homology extend this framework to complexes that lose simplices or evolve according to multiple parameters, respectively.Persistent-homology methods are widely used for higher-order real-data analyses, with examples discussed later in the paper.

3. Other measures of shape in simplicial complexes … 2. Combinatorial Laplacians

The paper surveys shape measures and higher-order Laplacians for simplicial complexes and hypergraphs, emphasizing their structural interpretations, limitations, and roles in analyzing diffusion and topology. Combinatorial Laplacians connect adjacent simplices, homology, spectral properties, and dynamical decompositions.

3. Other measures of shape in simplicial complexes: Homological invariants classify mesoscale structure but depend on the coefficient field, are not exhaustive, and may identify non-equivalent complexes as homologically indistinguishable.Despite these limitations, homology provides distinctive insights into dynamics in data spaces and has widespread applications.
3. Other measures of shape in simplicial complexes: The Euler characteristic counts simplices through the alternating sum χ = ∑D(−1)^k f_k, while spectral entropy measures simplex overlap through eigenvalues of L_k.Extensions of Laplacians to non-k-uniform hypergraphs require additional analysis.
E. Higher-order Laplacian operators: Higher-order Laplacians generalize graph Laplacians by assigning analogous roles to links, triangles, tetrahedra, and higher-dimensional simplices, but their definitions are not unique.A direct adjacency-based construction yields the Laplacian of a weighted graph associated with the chosen adjacency matrix [135, 136, 137].
E. Higher-order Laplacian operators: For k-uniform hypergraphs and k-regular simplicial complexes, adjacency tensors support Laplacian tensors discretizing higher-order Laplace–Beltrami operators.Their spectra can characterize diffusion, while other operators target specific diffusion processes [76] [78] or synchronization [141].
1. Hypergraph Laplacians: Hypergraph Laplacians include Chung’s construction for s-uniform hypergraphs and random-walk-based operators that represent k-walks through weighted undirected or Eulerian directed graphs.Lu and Peng also define α-lazy random k-walks, where the walker remains at the current edge with probability α.
2. Combinatorial Laplacians: For each simplex dimension k, combinatorial Laplacians combine lower and upper adjacency contributions encoded by boundary matrices and their adjoints.Orientations determine boundary matrices, whose products produce L_k; setting k = 0 recovers the standard graph Laplacian.
2. Combinatorial Laplacians: The kernel of L_k is isomorphic to H_k, so its dimension equals the number of homological k-holes; the Hodge decomposition separates globally acyclic, cyclic, and locally acyclic flows.For example, L_1 has a one-dimensional kernel corresponding to the toy complex’s one-dimensional hole.

IV. MODELS · A. Equilibrium models · 1. Bipartite models

The review organizes higher-order-system models into equilibrium and out-of-equilibrium stochastic processes, further grouped by representation while prioritizing higher-order interactions as first-class entities. Within equilibrium bipartite models, it covers configuration, exponential-random-graph, block, and latent-space approaches for generating structures and performing inference.

IV. MODELS: Equilibrium models define static distributions over higher-order systems, whereas out-of-equilibrium models define sequences of distributions and are respectively suited to inference or explaining emergent qualitative properties.The classification is useful despite formal correspondences between the two categories and between representations.
IV. MODELS: The review groups models by representation, highlights formal equivalences, and excludes models in which higher-order interactions arise only as byproducts of ordinary network structure.Examples excluded include models of non-trivial clustering and cliques.
1. Bipartite models: The bipartite configuration model controls interaction sizes and interactions per element through maximally random bipartite networks with fixed or expected degree sequences.Microcanonical versions impose exact degree constraints, whereas canonical versions fix expected degrees; these models support null modeling by testing which empirical properties follow from degrees alone.
1. Bipartite models: Bipartite configuration-model null ensembles showed that degree sequences can explain much of the structure in species co-occurrence, plant–pollinator, and Stack Overflow networks, though conclusions depend strongly on modeling assumptions [150, 151].Changing the model can substantially alter network-significance analyses [85].
1. Bipartite models: Bipartite ERGMs control motif frequencies to generate richer distributions, but degeneracy can concentrate probability on empty or fully connected networks and compromise sampling and inference.They are used both to generate networks matching observed motif counts and to infer structural-effect parameters, although subsampling can produce misleading inferences.
1. Bipartite models: Bipartite stochastic block models reproduce mesoscopic organization by assigning nodes in each part to blocks whose connection probabilities determine structure, with variants supporting degree correction, edge types, and hierarchy.By varying blocks and connection probabilities, these models can approximate arbitrary mesoscopic systems.
1. Bipartite models: Unlike configuration models, bipartite SBMs require inference of latent block assignments from network structure, linking their applications to topic modeling and data biclustering.Figure 7 illustrates joint and hierarchical block inference for people–events and words–texts.
1. Bipartite models: Latent-space models embed bipartite nodes in geometric or preference spaces, using distance and hidden variables to regulate connections, degree variation, and clustering.The AB random geometric graph uses a distance threshold, while S1 × S1 models combine circle distance with hidden variables; these methods infer latent geometry in metabolite–reaction networks.

2. Motifs models · 3. Stochastic set models

Motif models represent higher-order structure by assembling small graph patterns or constraining their distributions, while stochastic set models generate interactions through randomized group memberships and projections. Both frameworks extend network modeling beyond strictly pairwise structure, but motif selection and higher-order graph realization can be difficult.

2. Motifs models: Motif-based models assemble triangles, short loops, cliques, or other small graphs, thereby modeling higher-order relationships as distributions over classical networks.
2. Motifs models: ERGM-style motif models impose average constraints on network properties, including edges, closed triangles, and open triangles, for null modeling and motif-significance testing.
2. Motifs models: Configuration-based motif models preserve node degrees together with triangle, clique, or generic motif participation by matching typed stubs, including role- and node-type constraints.
2. Motifs models: General motif inference remains challenging because selecting a meaningful motif set is unclear, while dk-series models preserve local motif distributions to assess structural randomness but become difficult for k ≥3 [61] [62].
3. Stochastic set models: Stochastic set models randomize node–group memberships through bipartite configuration models, then project shared groups into node connections with probability q, representing hierarchical higher-order interactions.
3. Stochastic set models: Random intersection graphs are formally equivalent to projected bipartite models, with extensions allowing heterogeneous memberships, group sizes, hierarchies, mixtures, and noisy or probabilistic within-group connections.
3. Stochastic set models: These set-based models support structural, epidemiological, and clique-cover inference applications, whereas overlapping-community models are excluded because their communities are typically too large to represent interactions.

4. Hypergraphs models · 5. Simplicial complexes models

The paper reviews equilibrium models for hypergraphs and simplicial complexes, spanning random, multipartite, degree-controlled, community-based, latent-space, and Kronecker constructions. Simplicial-complex models additionally enforce inclusion among faces and support ER-like, geometric, exponential, configuration, and branching formulations.

4. Hypergraphs models: Hypergraph models generalize random-graph ideas to multi-body interactions, with Erdős–Rényi extensions assigning probabilities to hypergraphs with fixed or independently sampled hyperedges.The structural properties of these models, including component structure, have been extensively studied, and they support applications such as random k-SAT.
4. Hypergraphs models: Nonuniform hypergraph models allow several hyperedge sizes simultaneously, while k-partite models encode interactions among distinct node types, such as elements, people, and tags.Configuration variants control node degrees, and early applications modeled realistic folksonomies and node features [79].
4. Hypergraphs models: Hypergraph null models extend configuration, β-, and stochastic-block approaches to control degrees, node propensities, roles, weighted interactions, or community structure [81, 83, 85, 247, 248, 249, 250, 251, 252].These models are used especially for community-detection benchmarks and statistical estimation, with β-model probabilities increasing as node parameters β_i increase.
4. Hypergraphs models: Other hypergraph generators use latent geometry or Kronecker products to create realistic nested interactions and efficiently generate large random hypergraphs [253, 255].The latent-space construction uses multiple radii and random edge flips, allowing contained hyperedges and nonzero probability for every hypergraph [253].
5. Simplicial complexes models: Simplicial-complex models must distribute probability over facets while enforcing the inclusion of all lower-dimensional faces implied by each interaction.This distinction from hypergraphs matters especially when interactions have different sizes [102].
5. Simplicial complexes models: ER-like simplicial-complex models include Linial–Meshulam complexes, flag complexes derived from ER graphs, and the Δ-ensemble, whose parameter choices recover several established models.The Δ-ensemble and related formulations have been used for spreading, streaming-data prediction, and polymer studies.
5. Simplicial complexes models: Random geometric simplicial complexes connect randomly embedded points through Čech or Vietoris–Rips constructions, whose topology changes across sparse, thermodynamic, and diverging regimes of Λ = n r^d.For vanishing Λ complexes are disconnected and dust-like, constant Λ maximizes homology growth, and diverging Λ produces two higher-homology phase transitions; these models also support persistent-homology confidence intervals [267, 268].
5. Simplicial complexes models: Simplicial-complex configuration and branching models introduce realistic degree or dimension heterogeneity, randomize remaining structure, and support structural, null-model, and percolation analyses.Branching simplicial complexes attach randomly dimensioned faces iteratively to study percolation in a simplicial tree.

B. Out-of-equilibrium models … 4. Simplicial complexes models

Out-of-equilibrium models treat higher-order systems as evolving objects, typically using minimal, analytically tractable and mechanistically motivated rules to reproduce empirical structure. The review covers growth, rewiring and membership dynamics in bipartite networks, sets, hypergraphs and simplicial complexes.

B. Out-of-equilibrium models: Most out-of-equilibrium models are discrete-time growth processes that add nodes or edges without removal, often favoring existing nodes through degree-based preferential attachment.Their common goal is to reproduce empirical bipartite-network structure with minimal rules that support analytical calculations and plausible mechanisms.
1. Bipartite models: Bipartite models reproduce sexual, collaborative and competitive systems through preferential or uniform attachment, while alternatives model rearrangement, latent-space motion or rewiring.Preferential attachment concentrates degrees, whereas uniform attachment favors more equitable distributions; rewiring preserves the node set and can bridge dynamical and equilibrium descriptions.
2. Stochastic set models: Stochastic set models explain heavy-tailed memberships through rich-get-richer dynamics, including structural preferential attachment and hierarchical extensions.The Chinese restaurant and Indian Buffet processes provide related exchangeable models for single or multiple set memberships [289, 290].
3. Hypergraphs models: Hypergraph models grow folksonomies and co-authorship systems by combining intrinsic activity, preferential attachment and varying hyperedge rules [291–299].Their analysis has so far mainly evaluated whether node degree distributions are reproduced, typically alongside numerical simulations.
4. Simplicial complexes models: Simplicial-complex growth models such as Complex Quantum Network Manifolds and NGF generate geometries by adding facets, closing simplices and tuning saturation, flavor or energy parameters.The basic triangle model can produce planar graphs at low saturation bounds or complex geometries as the bound grows.
4. Simplicial complexes models: Other simplicial-complex models derive growing complexes from graph flag complexes, evolve manifold dimensions, or target dense directed complexes with scale-free degree distributions [305–308].These approaches respectively examine topological invariants and event sensitivity, classify manifold-evolution mechanisms, or reinforce and create weighted directed triangles.
4. Simplicial complexes models: Event-based dynamical simplicial-complex models create temporary simplices when nodes fire at activity-dependent rates, enabling study of aggregated structure and dynamics on continuously changing complexes [309].Each created simplex persists for a parameterized duration before disappearing.

V. DIFFUSION · A. Higher-order diffusion

The section distinguishes standard diffusion from continuous-time random walks and shows that both are governed by Laplacian spectra, with higher-order diffusion extending the dynamics from nodes to simplices. In simplicial complexes, diffusion depends on simplex order, producing order-specific spectra, relaxation, spectral dimensions, and return probabilities.

V. DIFFUSION: Standard diffusion moves concentration along network edges toward a conserved homogeneous consensus state under detailed balance.For undirected networks, the Laplacian’s zero-row-sum property conserves total concentration and makes the homogeneous state stationary.
V. DIFFUSION: In a connected network, standard diffusion converges to consensus at a rate set by the inverse of the smallest nonzero Laplacian eigenvalue, λ2.The zero eigenvalue is unique, while all other Laplacian eigenvalues are positive.
V. DIFFUSION: Continuous-time random walks differ because a single walker jumps between neighboring nodes, reaching a degree-proportional stationary distribution that favors hubs.The relaxation speed is likewise controlled by the second eigenvalue of the random-walk Laplacian.
V. DIFFUSION: Because standard diffusion is linear, extensions to higher-order interactions can be rewritten as pairwise dynamics with weights determined by higher-order organization.For three-body interactions, the higher-order tensor induces a new Laplacian matrix encoding these effective pairwise couplings.
V. DIFFUSION: To expose genuinely multi-body effects, higher-order diffusion must introduce nonlinearities; generalized Laplacians instead provide the framework for diffusion and random walks on edges, triangles, and tetrahedra.At higher orders, these simplices take the roles played by nodes in ordinary network Laplacians.
A. Higher-order diffusion: A simplicial complex supports distinct diffusive dynamics at each order k, with concentrations evolving on k-simplices and the resulting equations reducing to ordinary node diffusion when k = 0.For k ≥1, the diffusing substance occupies edges, triangles, or higher-order simplices, with Nk = |Xk| dynamical equations.
A. Higher-order diffusion: Higher-order diffusion on NGF simplicial complexes has spectral dimensions that increase with k, while spectral density is analytically related to simplex return-time probabilities.Figure 14 compares Laplacian spectra and return probabilities across k = 0, 1, 2, and 3, linking order-dependent spectra to dynamical behavior.

1. Edge-flows … 2. Random walks on hypergraphs

The section extends higher-order network analysis to edge-flow inference and random walks on simplicial complexes and hypergraphs. These models capture flow orientation, higher-dimensional adjacency, community-size effects, and interactions that cannot generally be reduced to projected graphs.

1. Edge-flows: Edge-based Laplacians encode flow orientation and support harmonic/gradient decomposition, denoising, smoothing, and approximate flow-conservation in applications such as vehicular traffic [51].Unlike vertex-based analysis, edge-based methods represent flow direction through signed entries and can process mass, energy, information, or traffic flows.
1. Edge-flows: Semi-supervised learning for edge-flows infers unlabeled divergence-free flows by selecting informative labeled edges and outperforms traditional alternatives on real-world street networks.The framework uses the edge-Laplacian to focus on approximately conserved flows, contrasting edge-based with vertex-based learning.
1. Random walks on simplicial complexes: Higher-order random walks can place walkers on edges or triangles, using lifted state spaces and orientation-sensitive transitions rather than ordinary node-to-node probabilities.Triangle walkers remain still with probability 1/2 or move across one of the current triangle’s edges to a lower-adjacent triangle.
2. Random walks on hypergraphs: Hypergraph random walks support arbitrarily large interactions; Zhou et al. use uniform hyperedge transitions and Laplacian eigenvectors to embed zoo animals for classification.The method selects a hyperedge incident to the current node, then chooses a node within it uniformly; the zoo hypergraph groups animals sharing features.
2. Random walks on hypergraphs: Size-weighted hypergraph walks spend more time in larger communities and can reverse node rankings relative to clique projections at intermediate hub sizes.For fixed hyperedge order k, the rankings agree when m < k + 1 or m > 1 + (k − 1)^2, but differ at intermediate m.
2. Random walks on hypergraphs: Edge-dependent vertex weights produce non-time-reversible hypergraph walks that cannot be reduced to traditional weighted-network walks, unlike edge-independent weighting.The model assigns distinct weights to vertices within the same hyperedge and yields analytically computable stationary states, with applications including scientific-collaboration ranking.
2. Random walks on hypergraphs: Nonlinear hypergraph Laplacians relate second-smallest eigenvalues to expansion, generalizing Cheeger’s inequality, while related frameworks extend analysis to diffusive, directed, and submodular hypergraphs [76] [78] [335].Higher-order random walks also support node ranking, community detection, topological data analysis, machine learning, quantum walks, and hypergraph cover-time theory [337].

VI. SYNCHRONIZATION · A. Phase oscillators · 1. Higher-order Kuramoto model

This section examines how higher-order interactions extend Kuramoto synchronization beyond pairwise coupling, focusing on how interaction order, topology, coupling functions, and spectral structure shape collective dynamics. It also considers multi-order Laplacian models, including attractive and repulsive couplings, and their implications for synchronization stability in simplicial systems and brain networks.

VI. SYNCHRONIZATION: Higher-order interactions can change synchronization transitions, favor cluster states, and produce new dynamical regimes beyond pairwise oscillator behavior.The section frames these effects within phase-oscillator synchronization, where coherence emerges from interactions among coupled dynamical systems.
1. Higher-order Kuramoto model: At fixed network size, motifs with more links require lower threshold coupling to synchronize, while delays induce multistability in unidirectional rings.The threshold is defined where synchronization probability exceeds 0.5; the delay result contrasts unidirectional with bidirectional rings.
1. Higher-order Kuramoto model: Pure three-body interactions produce abrupt desynchronization and infinitely many stable 2-cluster branches, each with a distinct transition threshold.The incoherent state remains stable when coupling increases, while coherent branches are distinguished by the relative cluster size η.
1. Higher-order Kuramoto model: In a maximal simplicial-complex model of macaque-brain connectivity, higher-order interactions induce explosive synchronization, hysteresis, bistability, and synchronization despite repulsive pairwise coupling.For the simplified all-to-all model, bistability and a hysteresis cycle occur above K2+3 = K2 + K3 = 2; sufficiently strong higher-order interactions stabilize synchrony when K1 < 0.
1. Higher-order Kuramoto model: The coupling function matters: symmetric and asymmetric three-body phase couplings are possible, but their dynamical implications have not yet been systematically investigated.Asymmetric forms can arise naturally from phase reduction of nonlinear oscillators, whereas symmetric forms generalize pairwise sinusoidal coupling.
1. Higher-order Kuramoto model: Higher-order oscillator formulations show that adaptive coupling between projected simplex dynamics makes synchronization explosive, while networks with spectral dimension dS > 4 are required for stable thermodynamic synchronization.For dS < 4, phase fluctuations diverge with network size and frustrated synchronization can occur; multi-order Laplacians provide a stability framework for identical-frequency oscillators.
1. Higher-order Kuramoto model: With attractive couplings at every order, increasing the order of pure interactions and the number of included orders makes synchronization more stable.For mixed attractive and repulsive couplings, interactions among orders can instead yield stability or instability, extending the result of Ref. [376].
1. Higher-order Kuramoto model: The multi-order Laplacian applies beyond all-to-all complexes, including simplicial star-clique models and real-world brain systems, and can probe other oscillatory dynamics.Decaying coupling strengths were also considered in analogy with higher-order phase-reduction techniques.
1. Higher-order Kuramoto model: Higher-order interactions stabilize synchronization: convergence is faster for 2-simplex than 1-simplex interactions, and stability increases with interaction order and maximum included order [141].The first non-zero Lyapunov exponent scales proportionally with order q, while the multi-order exponent becomes more negative as qmax increases.

2. Higher-order interactions from phase reduction

Higher-order phase reduction shows that many-body interactions arise naturally beyond first-order weak-coupling approximations, enabling dynamical regimes that pairwise phase models cannot capture. The reviewed theoretical, numerical, and experimental studies connect these interactions to clusters, chaos, heteroclinic switching, and other exotic behaviors.

Higher-order phase reduction: Higher-order weak-coupling terms naturally generate 2-, 3-, and 4-body interactions, extending phase models beyond Kuramoto-Sakaguchi dynamics and enabling cluster states, longer-time predictions, and multistability.These many-body terms can originate from cubic nonlinearities in the original oscillator system.
Higher-order phase reduction: Two nontrivial harmonics suffice for chaos in N = 4 when many-body interactions are included, whereas pure pairwise coupling requires at least four nontrivial harmonics.This result removes the previous need for higher harmonics or nontrivial amplitude dynamics to obtain chaos in small identical-oscillator networks.
Higher-order phase reduction: Nonpairwise interactions among oscillator populations produce heteroclinic connections and switching between weak chimeras with localized frequency synchrony.The existence and stability of these heteroclinic cycles were subsequently proved and analyzed.
Higher-order phase reduction: Second- and third-order isochron-based reduction of the mean-field complex Ginzburg-Landau equation reproduces its quasiperiodic partial synchronization and cluster dynamics.The higher-order phase model improves on first-order reduction, which yields the standard Kuramoto model with only synchronization and incoherence.
Higher-order phase reduction: For eight nano-electromechanical oscillators, second-order phase reduction generated three-body, next-nearest-neighbor, and biharmonic terms and qualitatively reproduced all observed complex regimes.Experiments observed weak chimeras, decoupled states, traveling waves, and inhomogeneous synchronized states.
Higher-order phase reduction: Analytical higher-order phase reduction can clarify exotic dynamics in general nonlinear oscillator networks, while numerical reduction offers an alternative when analytical derivations are unavailable.The section treats phase reduction theoretically and identifies future relevance for network inference.

B. Nonlinear oscillators … A. Spreading in higher-order networks

The section shows how higher-order interactions reshape synchronization, neuronal dynamics, network inference, and spreading processes beyond pairwise models. Across these settings, higher-order structure produces distinct dynamical regimes, improves or constrains inference, and can generate qualitatively different contagion behavior.

1. Chaotic oscillators: Stable synchronization in small chaotic-oscillator graphs generally coincides with evolutionarily preserved biological motifs, whereas the bifan motif cannot synchronize under any chaotic dynamics.Other studies show synchronization without oscillator correlations and oscillation death in fractal topologies.
1. Chaotic oscillators: Hypergraph and bipartite oscillator studies identify diverse regimes, including complete synchronization, oscillation death, partial anti-synchronization, cluster synchronization, and isolated desynchronization.Synchronization may require large hypergraph algebraic connectivity and passive linear coupling, while intra-population links can stabilize multisynchronous states.
2. Neuron models: Higher-order neuronal networks synchronize through adaptive feedback, motif-dependent pacemakers, anticipated synchronization, and resonance-pair mechanisms for zero-lag synchronization.Inhibitory-only motifs can exhibit up to eight coexisting attractors, while excitatory links promote synchronous bursting.
C. Inference of nonpairwise interactions in coupled oscillators: Phase-based inference reconstructs effective pairwise and higher-order couplings from scalar node time series by estimating phases, coupling functions, and partial norms.Effective links can exceed structural links because indirect terms appear at second order in the coupling strength, although structural and effective couplings are practically identical for sufficiently small ϵ.
C. Inference of nonpairwise interactions in coupled oscillators: Inference requires nonsynchronized, sufficiently long time series because phases must cover the N-torus; for N > 3, practical methods typically infer only partial pairwise dynamics and can create spurious links.Pairwise synchronization indices can miss triplet synchronization, and a large triplet index indicates possibility rather than sufficiency; triplets can synchronize while pairs do not.
C. Inference of nonpairwise interactions in coupled oscillators: Triplet partial analysis detects true links as well as pairwise analysis while avoiding the spurious links produced by pairwise inference in networks of 3 and 4 Van der Pol oscillators.Bayesian inference also detects pairs, triplets, and quadruplets under time-varying noisy conditions and outperforms pairwise-only inference, but remains limited to relatively small networks.
VII. SPREADING AND SOCIAL DYNAMICS: Social contagion models extend epidemic-style spreading to higher-order social contacts, where influence and peer pressure require complex-contagion mechanisms beyond simple disease transmission.The section distinguishes simplicial complexes from hypergraphs according to whether group interactions imply all sub-interactions.
A. Spreading in higher-order networks: In simplicial contagion, higher-order infection channels can create a discontinuous transition and bistability, with healthy and endemic states coexisting when λ∆ = 2.5 [262].The framework separates spreading on simplicial complexes from spreading on hypergraphs while allowing both higher-order and conventional dynamics [262].

1. Spreading on simplicial complexes … 2. Majority models

Higher-order spreading models show that simplicial sub-interactions, hypergraph group structure, and critical-mass mechanisms can qualitatively reshape contagion thresholds, transitions, and bistability. Related voter and majority dynamics demonstrate that higher-order updating and adaptive topology alter consensus, phase transitions, and critical behavior.

1. Spreading on simplicial complexes: Including 2-simplices changes contagion from a continuous to a discontinuous transition, with healthy and endemic states coexisting in a bistable region.The extended mean-field approach predicts the transition and its two basins of attraction on homogeneous social structures.
1. Spreading on simplicial complexes: Microscopic Markov-chain and link-equation approaches improve analytical predictions and extend contagion analysis to heterogeneous higher-order structures.
2. Spreading on hypergraphs: Hypergraph contagion separates group-only interactions from simplicial sub-interactions, while heterogeneous communities accelerate early spreading but can slightly reduce stationary prevalence.Global and local spreading strategies differ in long-term behavior: the global strategy has a vanishing epidemic threshold, whereas seed hyperdegree mainly affects early evolution.
2. Spreading on hypergraphs: On scale-free uniform hypergraphs, γ < γc = 2+1/(d−2) yields λc = 0, γ = γc gives a second-order transition, and larger γ produces hybrid transitions.Susceptibility remains finite below γc and diverges at the transition for higher γ, consistently with simulations.
2. Spreading on hypergraphs: Critical-mass thresholds in hypergraph contagion generate first- and second-order transitions and hysteresis, linking 10%–40% social-change thresholds to global and local group effects.The framework also introduces social latent heat as the fraction of individuals needed to move between dynamical solutions.
B. Opinion and cultural dynamics beyond pairwise interactions: Higher-order opinion models extend voter dynamics from pairwise copying to group-based updates, including hyperedge rules, coloring coordination, and adaptive simplicial peer pressure.Adaptive simplicial dynamics preserves active and frozen phases but increases convergence speed, lowers the critical rewiring threshold, and reduces active-edge density.
2. Majority models: Majority dynamics on regular hypergraph blocks reach consensus, with even group sizes favoring the bias and odd sizes producing growing clusters that eventually consolidate.This differs from high-dimensional voter dynamics, which can retain coexistence of both opinions.
2. Majority models: Noisy majority-vote models exhibit topology-dependent critical behavior: scale-free hypergraphs transition at finite qc even for 2 < γ < 3, with update rules affecting critical exponents differently.Hyperedge updates behave locally as mean-field, whereas node updates remain strongly shaped by heterogeneous topology.

3. Continuous models of opinion dynamics … 1. Public goods game

The section extends higher-order network dynamics from continuous opinions and cultural profiles to evolutionary games, emphasizing how group structure, topology, multiplexity, and coevolution shape cooperation and collective behavior.

3. Continuous models of opinion dynamics: Continuous opinion models represent states as xi ∈[0,1] and use nonlinear three-body interactions to combine peer pressure and homophily through influence functions.The influence function can make similar neighboring states accelerate or decelerate a third node’s dynamics.
3. Continuous models of opinion dynamics: Nonlinear higher-order interactions shift the asymptotic average opinion toward the initial majority when λ < 0 and toward balance when λ > 0, with modular topology adding local asymmetric effects.For λ = 0, the standard linear model conserves the average state.
4. Cultural dynamics: Cultural dynamics extend scalar opinions to vectorial or higher-order representations: Axelrod imitation depends on cultural overlap, while interconnected judgments can form simplicial complexes whose overlaps shape social interactions.Multiplex layers allow topic-specific interaction patterns and influence.
1. Public goods game: Public goods games generalize the prisoner’s dilemma to groups, where synergy and contribution rules determine payoffs through the reduced synergy factor r = R/G.Cooperators contribute a token while defectors contribute nothing; the multiplied pool is shared among all group members.
1. Public goods game: Network heterogeneity, community structure, and clustering can sustain cooperation at low synergy, while fixed-cost conventions determine whether scale-free networks strongly enhance network reciprocity [538, 549–554].Fixed cost per individual enhances cooperation more than fixed cost per game because highly connected cooperators obtain disproportionately large payoffs.
A. Multiplayer games on networks: Group interactions create indirect links and qualitatively distinct evolutionary dynamics, including self-criticality and indirect territorial competition, so local topology may be less decisive than in pairwise games.Multiplayer games can therefore produce collective outcomes that differ fundamentally from their pairwise counterparts.
1. Public goods game: Multiplex payoff coupling can enhance cooperation through interdependent network reciprocity and spontaneous symmetry breaking, but the benefit vanishes as interlayer edge overlap approaches zero.With zero overlap, cooperation requires every layer’s synergy factor to meet its isolated critical condition.
1. Public goods game: Coevolving interaction structures can increase social cohesion and prosocial behavior, while survival-based replacement may yield cooperation above the average degree and self-organized scale-free networks.Adaptive interlayer connections can also promote strong links around high-performing agents.

2. Other multiplayer games … A. Social systems

Higher-order interactions reshape evolutionary games by promoting cooperation, generating richer equilibria, and revealing structure-dependent dynamics. Applications to social systems show that hypergraphs and simplicial complexes capture affiliation, centrality, collaboration, friendship, and group evolution beyond pairwise networks.

2. Other multiplayer games: Multiplayer snowdrift, hawk-dove, and ultimatum games exhibit structure- and threshold-dependent outcomes, including enhanced prosociality, multiple equilibria, coexistence, bistability, and fairer offers.Dynamical grouping enhances prosocial behavior, while higher ultimatum acceptance thresholds produce more generous and fair outcomes.
1. Public goods game on bipartite networks: Cooperation is systematically enhanced on empirical bipartite collaboration networks compared with one-mode projections, because heterogeneous participation combines with homogeneous, often small, group sizes.This enhancement persists under both fixed cost per game and fixed cost per individual; larger groups reduce cooperation.
1. Public goods game on bipartite networks: Homogeneous group connectivity can produce more cooperation than scale-free substrates, while group overlap promotes cooperation similarly to clustering in projected networks.The result indicates that scale-free advantages depend on entangled social and group heterogeneities rather than solely fat-tailed game participation.
2. Public goods game on hypergraphs: Larger groups promote cooperation under lower synergy factors on sparse random hypergraphs, whereas increasing hypergraph density drives dynamics toward the well-mixed limit.The hypergraph formalism uses group hyperlinks and payoff normalization by played games, with strategy imitation depending on payoff differences.
PAYOFF COLLECTION: For heterogeneous hypergraphs, superlinear synergy breaks degeneracy, allowing hypergraph choice to independently set the cooperation threshold and relaxation time.When β ≠ 1, average relaxation time scales linearly with α; degeneracy persists for β ≤ 1.
IX. APPLICATIONS: Higher-order social applications span affiliation networks, urban groups, ritual participation, scientific collaborations, online networks, and other systems where interactions involve groups rather than dyads.Affiliation data can be represented as actor-event bipartite networks, hypergraphs, or dual hypergraphs.
A. Social systems: Hypergraph centrality assigns importance jointly to actors and events, and its application to 56 historical attacks captures more-than-dyadic participation among islands.The method extends Bonacich eigenvector centrality using the incidence matrix [108].
A. Social systems: Simplicial analyses show that social backcloth structure constrains friendships, q-connectivity can track group change, soccer q-holes obstruct play, and collaboration complexes exhibit strong simplicial closure.In Freeman’s study, 11 of 12 friendships were among 31 adjacent pairs, while Doreian’s temporal data were insufficient to explain the observed split.

B. Neuroscience and brain networks · C. Ecology

These sections examine how higher-order networks capture organization and dynamics beyond pairwise interactions in neuroscience and ecology, from neural activity and brain topology to ecosystem stability, biodiversity, host–pathogen–habitat associations, and species coexistence.

B. Neuroscience and brain networks: High-order correlations exist in neural populations, and higher-order interactions improve mesoscopic predictions of cortical dynamics, including neuronal avalanches in awake monkeys and visual responses in anesthetized cats.Methods measure multi-spike interaction strengths and show these interactions shape cortical-column dynamics and matter for population coding.
B. Neuroscience and brain networks: Hippocampal correlation and coactivity complexes encode spatial topology, with real Betti curves lower than randomized models and temporal activity compensating for network deterioration to preserve map stability.Accumulated coactivity approximates the topology of the animal’s environment, while compensatory neuronal activity maintains a consistent representation despite remapping and fluctuating population coding.
B. Neuroscience and brain networks: Topological methods identify biologically meaningful brain organization and distinguish healthy, pathological, developmental, neurodegenerative, cognitive, and altered-consciousness states [598, 625, 634–644].White-matter cliques cluster around reproducible cavities that guide information flow, while psilocybin-associated scaffolds reveal different patterns of regional information integration.
C. Ecology: Hypergraphs represent trait-mediated indirect interactions and their modifications, enabling ecological analyses of shortest hyperpaths and centrality that pairwise networks can fail to capture.In the coffee agroecosystem, ant protection of crops was modified by ant-parasitizing phorid flies, and these higher-order effects were crucial for controlling agricultural pests.
C. Ecology: Higher-order ecological interactions can stabilize coexistence: four-species interaction thresholds increase with diversity, whereas pairwise thresholds decrease as 1/N, producing lower and upper diversity bounds.In mixed systems, increasing total interaction strength narrows the allowed diversity range, and some intermediate ecosystem sizes can be stable while smaller and larger systems are unstable.
C. Ecology: Adding higher-order competition changes unstable pairwise cycling into globally stable coexistence, delays stochastic extinctions, and makes species coexistence robust to perturbations.Sampling three seedlings instead of two preserves the pairwise model’s equilibrium while converting cycles into a globally stable fixed point; fourth- and higher-order terms accelerate convergence.
C. Ecology: A HOrS framework represents plant–virus–habitat associations as hypergraph hyperlinks, enabling host-ecotype interactions to be studied with explicit spatial context.The framework was applied to environmentally mediated host–pathogen infections.
C. Ecology: A neutral multi-habitat evolution model found that real ecosystems lie on a continuum between nested and modular networks, validated across agricultural ecosystems in Spain.This goes beyond treating modularity and nestedness as a strict dichotomy.
C. Ecology: Levine et al. review how non-pairwise interactions contribute to maintaining biodiversity and species coexistence in complex ecological communities.The cited review provides a broader account of these ecological mechanisms.

D. Other biological systems · X. OUTLOOK AND CONCLUSIONS

Higher-order representations reveal biological structure and dynamics that pairwise models can miss, improving functional prediction, robustness analysis, drug-combination discovery, and disease-association prediction. The field’s outlook centers on developing measures, generative models, dynamical theory, and inference schemes for higher-order systems.

D. Other biological systems: Hypergraph models improve functional predictions for yeast and human protein pathways and reveal protein-complex properties obscured by pairwise projections.Hypergraph degree correlates better with gene essentiality than graph degree, while larger complexes tend to be more essential.
D. Other biological systems: Higher-order metabolic models show that environmental variability increases robustness, while tensor spectra and combinatorial Laplacians capture chemical and reaction-network information.Metabolic hypergraphs exhibit core-periphery structure, and reactions require all metabolites in a hyperedge to be active under site percolation.
D. Other biological systems: Dose-based models predict effective drug combinations of up to ten drugs, but pairwise inference becomes less efficient with noise and can be less noise-resistant than the pairs model.The dose model was tested on combinations involving E. coli and M. tuberculosis pathogens, while the pairs model trades precision for noise resistance.
D. Other biological systems: Higher-order dynamics and inference improve biological prediction, including disease-microbe associations, dynamic gene-metabolite correlations, and reconstruction of incomplete cellular interaction data.Higher-order random walks outperform traditional random walks for disease-microbe association prediction, while higher-order inference captures global dynamic correlation patterns.
X. OUTLOOK AND CONCLUSIONS: Higher-order systems differ structurally and dynamically from pairwise models, introducing new nonlinearities and allowing state variables on edges, triangles, and higher-dimensional objects.These representations motivate concepts such as group states, whose meaning and interpretation remain unresolved.
X. OUTLOOK AND CONCLUSIONS: Higher-order structure can alter critical dynamics even without explicit higher-order dynamical terms, producing mesoscopic localization, outbreak persistence, and shifts in critical infectivity.For temporally evolving contacts, the critical-infectivity shift depends on a trade-off between group-size distributions and node-activity distributions.
X. OUTLOOK AND CONCLUSIONS: Open methodological needs include measures for temporal, multiplex, mesoscopic, simplicial, and homological structure, plus generative models reproducing refined topology, temporal structure, and multiplex interactions.Existing simplicial-complex models generally reproduce local connectivity but not target homology or mesoscopic structures, and most higher-order models focus on growth.
X. OUTLOOK AND CONCLUSIONS: The field still lacks general mechanisms for higher-order dynamics and robust inference schemes distinguishing genuine higher-order interactions from low-order effects, especially when data are sparse or nonmechanistic.Broader applications are expected in biology, ecology, population dynamics, neuroscience, and computational social science.