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What are higher-order networks?
Christian Bick, Elizabeth Gross, Heather A. Harrington, Michael T. Schaub
TL;DR
Graphs provide simple and useful models but cannot directly represent nondyadic relationships, motivating higher-order networks. This survey integrates higher-order representations across data geometry, relational modeling, and network dynamics, reviewing their mathematical tools and applications. It concludes that these representations can preserve richer structure, while their interpretation may depend on choices such as filtrations and coordinate systems.
Problem
Graph-based models encode only dyadic relationships, whereas many systems involve polyadic interactions that require higher-order representations.
Method
The paper surveys and integrates higher-order networks across topology and geometry of data, statistical modeling of relational data, and higher-order network dynamics.
Results
Higher-order networks support richer data-geometric and topological analysis, direct modeling of nondyadic relations, and dynamical systems with joint nonlinear interactions.
Takeaways & Limitations
Higher-order representations provide a common framework for analyzing geometry, relational structure, and dynamics beyond pairwise graph models.
Takeaways & Limitations
Higher-order dynamical representations may depend on coordinate choices, and persistent homology requires an application-dependent filtration.
Abstract
from arXiv · showhide
Network-based modeling of complex systems and data using the language of graphs has become an essential topic across a range of different disciplines. Arguably, this graph-based perspective derives its success from the relative simplicity of graphs: A graph consists of nothing more than a set of vertices and a set of edges, describing relationships between pairs of such vertices. This simple combinatorial structure makes graphs interpretable and flexible modeling tools. The simplicity of graphs as system models, however, has been scrutinized in the literature recently. Specifically, it has been argued from a variety of different angles that there is a need for higher-order networks, which go beyond the paradigm of modeling pairwise relationships, as encapsulated by graphs. In this survey article we take stock of these recent developments. Our goals are to clarify (i) what higher-order networks are, (ii) why these are interesting objects of study, and (iii) how they can be used in applications.
1. Introduction.
Higher-order networks extend graph-based modeling beyond pairwise relationships to capture polyadic interactions and richer geometric, topological, relational, and dynamical structure. This survey integrates these perspectives and reviews mathematical tools and applications across three contexts.
- Motivation: Graphs represent entities as vertices and interactions as pairwise edges, limiting graph-based models to dyadic relationships.In network dynamics, this means an interaction between two nodes is independent of other nodes.
- Motivation: Higher-order networks capture joint interactions among more than two nodes, including group coordination and mult species biochemical reactions.The paper groups supra-dyadic, nonpairwise, higher-order, and simplicial interactions under this term.
- Survey scope: The survey develops an integrated perspective on higher-order networks and reviews recent progress across differing research motivations, questions, and mathematical tools.It aims to provide a common language for research areas often considered separately.
- Topology and geometry of data: For data geometry and topology, simplicial complexes generalize pairwise geometric relationships to local n-wise relationships and retain more information about data geometry.They support persistent homology and hierarchies of Hodge-Laplacians, with the graph Laplacian as a special case.
- Analysing and modelling relational data: For relational data, higher-order models preserve observed nondyadic relationships instead of preprocessing each set into pairwise subsets.Prominent representations include hypergraphs and abstract simplicial complexes, which can be analyzed with probabilistic models.
- Higher-order network dynamical systems: Higher-order network dynamical systems allow joint nonlinear interactions among three or more oscillators, while coordinate transformations can create apparent nondyadic interactions.Such representations are not necessarily invariant under coordinate transformations.
2. A brief review of graphs, hypergraphs, and simplicial complexes.
Higher-order networks use hypergraphs and simplicial complexes to represent relationships involving more than two vertices. Hypergraphs allow arbitrary vertex subsets as hyperedges, while simplicial complexes additionally require closure under inclusion.
- Higher-order network objects: Hypergraphs and simplicial complexes encode higher-order interactions beyond pairwise graph edges.A hypergraph generalizes a graph by allowing hyperedges that are arbitrary non-empty vertex subsets; simplicial complexes are a special closed-under-inclusion case.
- Graphs: An undirected graph represents relationships as unordered pairs of vertices, whereas a directed graph uses ordered pairs.
- Hypergraphs: A k-uniform hypergraph has every hyperedge containing exactly k vertices.The section’s example is a 3-uniform hypergraph with 9 vertices and 6 edges.
- Simplicial complexes: A simplicial complex contains every non-empty subset of each simplex, making its 1-skeleton the associated graph.
- Simplicial complexes: Orientations order simplex vertices and distinguish even from odd permutations, extending directional structure beyond ordinary edges.An oriented 1-simplex has a direction, while an oriented 2-simplex has a sense of rotation.
- Applications: Topological properties of simplicial complexes, including connected components and loops, can describe the shape of datasets.Homology provides a more detailed account of topology than the Euler characteristic.
3. Geometry and topology of data with (higher-order) networks.
Simplicial homology studies cycles and boundaries across dimensions to quantify topological features, while persistent homology tracks these features across scales. These tools support higher-order data analysis, but applications depend on choosing a suitable filtration.
- Simplicial homology: Chain groups are generated by oriented simplices, and boundary operators map each simplex to an alternating sum of its faces.Coefficients lie in a field, and reversing orientation changes the corresponding coefficient’s sign.
- Simplicial homology: The composition of consecutive boundary maps is zero, so every boundary is a cycle and the chain groups form a chain complex.Formally, ∂n ◦∂n+1 = 0 and Bn ⊆ Zn.
- Simplicial homology: Homology quantifies cycles that are not boundaries, interpreting them as n-dimensional holes.The first Betti numbers correspond to connected components, one-dimensional holes, and two-dimensional holes.
- Persistent homology: Persistent homology studies homology across a filtration of nested simplicial complexes, tracking when features appear and disappear across scales.Persistent features can indicate structure such as clustered data, and the resulting barcode is robust to small input perturbations.
- Applications of persistent homology: A suitable filtration must be chosen for point-cloud data, and that choice depends strongly on the application.The reviewed applications focus on simplicial complexes or their 1-skeletons rather than general hypergraph filtrations.
- Applications of persistent homology: Higher-order and topological methods have been applied across neuroscience data, including functional networks, neuronal morphology, synaptic connectivity, place cells, connectomes, and brain disease imaging.
4. Understanding relational data with higher-order network models.
Higher-order models represent relational data without reducing nondyadic relationships to pairs, but the chosen abstraction determines which statistical and topological tools are available. The section surveys probabilistic models ranging from random hypergraphs to configuration models.
- Representing relational data: Higher-order models preserve measured nondyadic relationships instead of preprocessing each relation into size-two subsets.This avoids discarding the original relational structure, while still requiring choices about how higher-order relations are represented.
- Representing relational data: Simplicial complexes include every subset of each modeled relation, an assumption that may fit author collaborations but not enzyme-mediated reactions.For a collaboration among authors A, B, and C, the model treats each subset as present; biochemical examples can violate this implication.
- Representing relational data: The representation strongly affects downstream analysis: simplicial complexes support applied-topology tools, whereas hypergraphs are more flexible but have fewer developed tools.Available analyses include questions about homology, such as whether gaps occur in an author-collaboration space.
- Probabilistic models: Statistical analysis generally requires a probabilistic model over the selected higher-order representation, beyond merely describing the data.The section introduces statistical models as collections of probability distributions and uses Erdős–Rényi models as a starting point for higher-order generalizations.
- Probabilistic models: The hypergraph model H_d(N,p) selects each d-uniform hyperedge independently with probability p; for d=3 and N=4, its sample space contains 16 hypergraphs.Figure 5 projects the one-dimensional H_3(4,p) model onto three coordinates.
- Probabilistic models: Random geometric complex models serve as null models for statistics including Euler characteristics, homology, and persistence-diagram summaries.These models support analysis of statistics generated from simplicial complexes built from random geometric data.
- Probabilistic models: Hypergraph configuration models have been used as null models for tripartite photo-sharing data and collaboration or communication networks, including annotated hypergraph extensions.Annotated hypergraphs generalize directed hypergraphs by assigning each vertex a role within each edge.
5. Network Dynamical Systems with Higher-Order Interactions.
The section shifts from using network-generated data to studying general interacting dynamical systems directly. It begins with the conventional graph-based formulation of network dynamics.
- Overview: Network dynamical systems can generate trajectory data for methods discussed earlier, but this section focuses on general properties of interacting dynamical systems.The emphasis is on a dynamical-systems perspective rather than only analyzing sampled network data.
- Overview: Traditional network interactions are encoded by a directed, weighted graph with N vertices and a weighted adjacency matrix A.The setup assumes identical nodes and introduces graph-based coupling as the baseline for later higher-order extensions.
5.1. Network dynamical systems with pairwise interactions.
Pairwise network dynamics combine intrinsic node evolution with coupling functions acting on pairs of connected nodes. Their standard formulation is nonlinear in states but additive in coupling weights.
- Pairwise interactions: Each node follows intrinsic dynamics ˙x_k=F(x_k), while connected node pairs interact through a coupling function G.The state x_k belongs to R^d, and G maps two node states to R^d.
- Pairwise interactions: The collective state evolution depends on the graph structure, the intrinsic dynamics F, and the pairwise coupling function G.The graph structure is encoded by the adjacency matrix A.
- Pairwise interactions: Although interactions may be nonlinear in node states, the standard model is linear in coupling weights A_kj, making graph changes adjustable through those weights.This additive interaction structure is a central property of the conventional formulation.
- Pairwise interactions: For general systems, node-state evolution need not be expressible as a superposition of pairwise interactions.The broader formulation allows each node’s dynamics to depend on the joint state of the entire network.
5.2. Network dynamical systems with higher-order interactions.
Higher-order dynamical systems allow node evolution to depend concurrently on multiple node states, motivating higher-order structures beyond adjacency matrices. However, the representation of a generically coupled system is not unique and can depend on perspective.
- Higher-order interactions: Each function H_k can depend on the joint state of all N nodes, allowing interactions that involve multiple nodes concurrently.This generalizes pairwise coupling by permitting the influence on node k to depend on more than two node states.
- Higher-order interactions: Graph-based node dependencies do not exclude nonlinear interactions involving multiple nodes, as illustrated by coupled cell systems.A graph can encode which nodes affect a node even when the compatible function depends jointly on several inputs.
- Higher-order interactions: A formal expansion separates pairwise terms from nonpairwise terms, with higher-order interactions represented by coefficients A^(s) and coupling functions G^(s) for s≥3.For example, terms involving x_k, x_j, and x_l describe the joint influence of nodes j and l on node k.
- Higher-order interactions: Higher-order dynamics raise questions about their dynamical consequences, their necessity for real systems, and the appropriate representing structure.Candidate structures include hypergraphs and simplicial complexes.
- Higher-order interactions: The higher-order expansion of a generically coupled system is generally nonunique, so questions about whether higher-order interactions are crucial and which structure is appropriate may lack well-defined answers.The authors frame structure choice as potentially a matter of perspective.
5.3. Effects of higher-order interactions in network dynamical systems.
Higher-order interactions alter collective dynamics across oscillator, ecological, neural, and discrete-state network models. These effects include changes to synchronization, coexistence, oscillation, memory, and contagion behavior.
- Phase oscillator networks: Higher-order terms in Kuramoto-type models can affect synchronization and produce multistability, heteroclinic cycles, chaotic dynamics, and explosive synchronization.They also arise in physical systems including electronic circuits, nanomechanical oscillators, and optical devices.
- Ecological networks: Higher-order interactions in competitive population models can make species coexistence robust to perturbations in abundances and parameter values.
- Neural networks: Nonadditive neural interactions support synchronous activity, high-frequency oscillations, and memory processes.The model uses nonlinear amplification and saturation of joint inputs to neurons.
- Other network dynamical systems: Higher-order interactions also arise in discrete-time or discrete-state systems, including contagion dynamics on simplices and adaptive voter models with rewiring.
5.4. Algebraic structures for network dynamics with higher-order in-
The relation between higher-order network structure and dynamical interactions depends on representation and reduction. Higher-order terms can emerge from phase reductions of pairwise systems, while recovering algebraic structure from dynamics is generally nontrivial.
- Network dynamics on hypergraphs and simplicial complexes: A hypergraph can define network dynamics, but associating an algebraic structure with a given dynamical system depends on interaction coefficients, coupling functions, and symmetry assumptions.
- Asymptotic expansions and phase reduction: Phase reduction approximates weakly coupled oscillator dynamics on an attracting invariant torus and generically produces higher-order interactions.Explicit calculations near a supercritical Hopf bifurcation find nonpairwise phase coupling already at first order in ε.
- Effective coupling: Additively coupled nonlinear oscillators can yield nonadditive phase dynamics after reduction, including higher harmonics and triplet interactions.
- Effective coupling: Reducing graph-based dynamics to a lower-dimensional invariant manifold produces a more complicated coupling structure that encodes nearby vector-field nonlinearity.This can relate dynamical systems with and without higher-order interactions through coordinate changes.
- Modeling perspective: Higher-order network tools create opportunities to choose modeling representations more critically when data are given as matrices.
6. Discussion.
Higher-order networks are presented as an emerging field whose previously separate aspects are interlinked. The review advocates cross-fertilization through an integrated perspective connecting these research directions.
- 6. Discussion.: Structural properties and dynamical behavior can be studied together through structure–function relationships in neuroscience.The paper uses neural structural networks and their dynamical systems as an example of the field’s interlinked aspects.
- 6. Discussion.: Higher-order networks open new research directions despite appearing to draw on results from multiple fields.The authors frame this apparent patchwork as an opportunity rather than a barrier.
- 6. Discussion.: An integrated review and perspective is offered as a step toward cross-fertilization across these research areas.The paper positions integration as a way to connect developments that have often been studied separately.