Source-linked AI summary
Signal Processing on Higher-Order Networks: Livin' on the Edge ... and Beyond
Michael T. Schaub, Yu Zhu, Jean-Baptiste Seby, T. Mitchell Roddenberry, Santiago Segarra
TL;DR
Signal processing on higher-order networks lacks the mature theoretical foundation available for graphs, despite widespread polyadic interactions. This tutorial develops processing tools for simplicial complexes and hypergraphs, covering spectral analysis and applications while comparing representation choices and their limitations.
Problem
Polyadic interactions require higher-order network models, but signal-processing methods for these structures remain comparatively sparse and their appropriate algebraic operators are not established.
Method
The tutorial develops signal-processing foundations for simplicial complexes and hypergraphs, using Hodge-Laplacian, matrix, and tensor representations across several processing tasks.
Results
SSL is 0.99 for the presented interpolation and semi-supervised learning example, with a 2-norm error of 0.064.
Takeaways & Limitations
Choosing the higher-order representation and shift operator determines the associated notion of signal smoothness and shapes the resulting processing framework.
Takeaways & Limitations
Interleaving nonlinearities enables extended simplicial locality but removes the boundary-operator composition structure from joint input-output relations.
Abstract
from arXiv · showhide
In this tutorial, we provide a didactic treatment of the emerging topic of signal processing on higher-order networks. Drawing analogies from discrete and graph signal processing, we introduce the building blocks for processing data on simplicial complexes and hypergraphs, two common higher-order network abstractions that can incorporate polyadic relationships. We provide brief introductions to simplicial complexes and hypergraphs, with a special emphasis on the concepts needed for the processing of signals supported on these structures. Specifically, we discuss Fourier analysis, signal denoising, signal interpolation, node embeddings, and nonlinear processing through neural networks, using these two higher-order network models. In the context of simplicial complexes, we specifically focus on signal processing using the Hodge Laplacian matrix, a multi-relational operator that leverages the special structure of simplicial complexes and generalizes desirable properties of the Laplacian matrix in graph signal processing. For hypergraphs, we present both matrix and tensor representations, and discuss the trade-offs in adopting one or the other. We also highlight limitations and potential research avenues, both to inform practitioners and to motivate the contribution of new researchers to the area.
1. Introduction
Graphs model pairwise interactions, but many systems involve polyadic relations that require higher-order abstractions. This tutorial addresses the comparatively sparse literature on processing signals over such networks and the challenges of choosing suitable operators.
- Graphs encode entities as nodes and pairwise interactions as edges across applications including neuroscience, transportation, and social science.
- Graph signal processing provides a framework for transforms, filtering, sampling, and reconstruction of data supported on graphs.
- Higher-order models such as simplicial complexes and hypergraphs represent polyadic interactions that graphs cannot capture.
- Signal processing for higher-order networks remains comparatively sparse relative to work on representing their multi-relational structure.
- Simplicial complexes and hypergraphs have different modeling trade-offs, with hypergraphs offering greater flexibility and simplicial complexes imposing structured allowed hyperedges.
- Higher-order analysis is challenged by combinatorially many interactions, large matrix and tensor representations, computational and statistical issues, and an underdeveloped theory.
2. Signal processing on graphs: A selective overview
The tutorial first revisits graph and discrete signal-processing principles to guide its treatment of signal processing on higher-order networks, focusing on undirected structures.
- The tutorial focuses on undirected graphs and higher-order networks while noting that directed-graph signal processing has also been studied.
2.1. Central tenets of discrete signal processing
Discrete signal processing represents filtering through operators and interprets shift-invariant filters spectrally. Graph Fourier decomposition analogously uses Laplacian eigenvectors ordered by signal smoothness.
- A linear filter maps an input signal to a transformed output through the matrix-vector operation s_out = Hs_in.
- Shift-invariant filters commute with the shift operator, preserving each other’s eigenspaces and enabling Fourier-based interpretation.
- Graph Laplacian eigenvectors form a basis ordered from low to high graph frequencies as their smoothness decreases.
- Decomposing a node signal into Laplacian eigenvectors yields its Fourier coefficients.
2.2. Graphs, incidence matrices, and the graph Laplacian
An undirected graph consists of nodes and unordered pairwise edges. Its incidence matrix provides an oriented algebraic representation, yielding the equivalent Laplacian expression L = BB^⊤.
- An undirected graph contains nodes and edges formed by unordered pairs of nodes.
- The symmetric adjacency matrix records whether each node pair is connected, while the degree matrix is formed from row sums of the adjacency matrix.
- Assigning each edge an orientation defines an incidence matrix with −1 and +1 entries at its tail and head nodes.
- The graph Laplacian is equivalently represented as L = BB^⊤ using the incidence matrix.
2.3. Graph signal processing
Graph signal processing represents node data as vectors and processes it with relational shift operators, especially the graph Laplacian, whose spectrum supplies a Fourier basis ordered by signal variation.
- A graph signal assigns a real value to each node and can be represented as a vector in R^N.
- Filtering applies a matrix operator to an input signal, using relational operators such as the adjacency matrix or graph Laplacian.
- The eigenvectors of a shift operator define the graph Fourier transform, while its eigenvalues determine the filter’s frequency response.
- Graph filtering projects a signal into the Fourier basis, scales frequency modes, and transforms the result back to the node domain.
- The positive-semidefinite graph Laplacian provides non-negative frequencies, with low and high eigenvalues corresponding to small and large variation across graph edges.
2.4. Graph signal processing: Illustrative problems and applications
The tutorial uses graph signal processing to frame denoising, interpolation, node embeddings, and graph neural networks as spectral or smoothness-based operations on graph-supported data.
- Illustrative problems and applications: Graph signal processing has been applied to sampling, reconstruction, deconvolution, and network topology inference.
- Node embeddings: Spectral node embeddings use truncated Fourier representations of node indicator vectors, while high-frequency eigenvectors can reveal nearly bipartite node sets.
- Signal denoising: Graph denoising assumes a smooth underlying signal and balances Laplacian-based smoothness against squared error to observations.
- Signal denoising: Iterative smoothing performs gradient-descent steps on the smoothness cost, and both iterative and optimization-based procedures act as low-pass filters.
- Signal interpolation: Signal interpolation assigns values to unlabeled nodes by minimizing connected-node label differences while preserving observed labels.
- Graph neural networks: Graph neural networks combine graph filtering, permutation invariance, Fourier analysis, nonlinear activations, and learnable feature transformations for embedding and classification tasks.
- Graph neural networks: With identity activation and scalar features, the architecture reduces to iterative graph smoothing; nonlinear activations and learned weights enable more sophisticated node-feature relationships.
3. Modeling higher-order interactions with simplicial complexes
Simplicial complexes extend graph structure to higher-order simplices, enabling signals on nodes, edges, and triangles and motivating the Hodge Laplacian as a natural shift operator.
- Hodge Laplacian: The Hodge Laplacian generalizes the graph Laplacian and provides an orthogonal spectral basis related to Hodge decomposition.
- Simplicial complexes: A simplicial complex contains simplices together with all their subsets, so each simplex has faces and may belong to higher-dimensional co-faces.
- Simplicial complexes: Orienting simplices by ordered vertices supports boundary operators that record relationships between consecutive simplex dimensions.
- Signals on simplices: Signals may be defined on k-simplices, including nodes, edges, and triangles, as illustrated by the running simplicial-complex example.
- Hodge Laplacian: For k=0, the Hodge Laplacian recovers the graph Laplacian, while weighted variants include random-walk and normalized graph Laplacians; the tutorial focuses on the unweighted case.
- Hodge Laplacian: For oriented edge-flow signals, the Hodge Laplacian is relevant because it incorporates the chosen edge orientations and higher-order structure.
- Hodge Laplacian: The positive-semidefinite Hodge Laplacian yields non-negative frequencies whose eigenvectors encode smoothness through the Hodge decomposition.
1 B1+ B2B⊤
Hodge decomposition separates edge flows into orthogonal gradient, curl, and harmonic components, with the Hodge 1-Laplacian’s eigenvectors spanning the gradient and curl subspaces.
- Gradient flows: Gradient flows arise from differences of node potentials and therefore cannot produce positive net flow around a closed path.
- Curl and harmonic flows: Curl flows are local circulations generated by assigning potentials to 2-simplices and applying the boundary operator.
- Curl and harmonic flows: The harmonic subspace is the kernel of the Hodge 1-Laplacian, while harmonic and curl flows together form the cycle space.
- Worked example: In the example, the curl component decomposes into two local circulations with absolute magnitudes 1 and 1.7.
- Spectral decomposition: Nonzero Hodge 1-Laplacian eigenvectors split into groups spanning the gradient and curl spaces, with graph-Laplacian and 2-simplex coupling spectra supplying them.
- Spectral decomposition: Edge flows can contain high-frequency contributions from either curl components or strong gradient components, unlike node signals’ single corresponding basis-component interpretation.
4. Signal processing and learning on simplicial complexes
The section extends signal processing from graphs to simplicial complexes using Hodge-Laplacian-based edge and trajectory embeddings, smoothing, denoising, and interpolation. These methods exploit harmonic structure, flow conservation, and curl to capture topology and improve signal processing.
- 4.1. Fourier analysis: Edge-flow and trajectory embeddings: Hodge-Laplacian eigenvectors provide low-frequency edge embeddings, with harmonic embeddings projecting edge signals into the zero-frequency harmonic subspace.The harmonic space corresponds to globally cyclic flows associated with the complex’s 1-homology, or unfilled holes.
- 4.1. Fourier analysis: Edge-flow and trajectory embeddings: Trajectory vectors encode oriented edge traversal, and their harmonic projections can be accumulated sequentially or computed directly from the full trajectory vector.Linearity makes the sequential and direct projections produce the same final embedding.
- 4.1. Fourier analysis: Edge-flow and trajectory embeddings: Trajectories with similar behavior around simplicial-complex holes obtain similar harmonic embeddings, enabling identification of topologically similar trajectories by clustering.In the example, trajectories passing above the same obstacle or between the two obstacles cluster similarly, while the trajectory taking a distinct route stands out.
- 4.2. Flow smoothing and denoising: The edge Laplacian promotes flow conservation by assigning zero regularization cost to cyclic flows and penalizing flows with nonzero divergence.Its quadratic form is based on the incidence matrix and therefore vanishes on the graph cycle space.
- 4.2. Flow smoothing and denoising: The Hodge 1-Laplacian refines smoothness by adding a squared curl penalty, allowing cyclic flows with curl to have nonzero frequencies.Including 2-simplices provides modeling flexibility for shaping an edge-flow filter’s frequency response.
Simplicial Graph Neural Networks
Simplicial graph neural networks extend graph-neural-network ideas to signals on multiple simplex levels while enforcing orientation and locality properties. Nonlinear activations enable information from all levels to influence each output, but this trades away some boundary-operator structure.
- Simplicial Graph Neural Networks: The architectures process node, edge, and face signals using boundary operators, coboundary operators, Hodge Laplacians, aggregation functions, and nonlinear activations.Their design targets permutation invariance, orientation invariance, and simplicial locality.
- Simplicial Graph Neural Networks: Orientation equivariance requires outputs to transform consistently when the arbitrary reference orientations of simplices change.For flow architectures based on the Hodge Laplacian, an odd activation function ensures compatibility with orientation changes.
- Simplicial Graph Neural Networks: Simplicial locality restricts each layer’s k-simplex output to inputs from adjacent levels k−1, k, and k+1.This locality condition concerns information exchange between neighboring levels of the simplicial complex.
- Simplicial Graph Neural Networks: With linear activations, repeated boundary-operator layers remain unable to incorporate information from nonadjacent simplex levels.The limitation follows from the null composition of boundary operators.
- Simplicial Graph Neural Networks: Nonlinear activations produce extended simplicial locality, allowing each output level to depend on inputs from all simplex levels.This global property can coexist with layerwise simplicial locality.
- Simplicial Graph Neural Networks: Extended simplicial locality trades full cross-level influence for loss of the boundary-operator composition structure in joint input-output relationships.Although Hodge decomposition remains applicable to outputs, the strict upper- and lower-incidence characterization no longer holds jointly.
5. Modeling higher-order interactions via hypergraphs
Hypergraphs provide flexible representations of polyadic interactions that may not imply all pairwise or subset interactions, while matrix- and tensor-based operators offer different processing trade-offs.
- Hypergraph motivation: Compared with simplicial complexes, hypergraphs avoid the requirement that every included interaction also contain all recursively included faces.This flexibility also supports heterogeneous weighting and other directionality patterns that may not fit simplicial-complex structure.
- Hypergraph motivation: Hypergraphs model group interactions directly, allowing relationships among more than two vertices and distinguishing exclusive multi-node interactions from implied pairwise links.In co-authorship, a paper with three or more authors need not imply that those authors collaborated in pairs.
- Hypergraph models: Hypergraphs can be extended to heterogeneous vertices or hyperedges, edge-dependent vertex weights, and inhomogeneous hyperedges with cut-dependent costs.These variants represent multiple entity or relation types, unequal contributions within a group, or different costs for partitioning a hyperedge.
- Shift operators: Signal processing on hypergraphs requires choosing a shift operator, with matrix-based and tensor-based representations providing the two major approaches.Matrix methods derive graph-like operators from a matrix representation, whereas tensors offer a multidimensional representation with a richer set of possible shift operators.
- Open issues: The relative advantages and disadvantages of the many matrix constructions remain sparsely understood, making spectral analysis important for selecting representations.Each matrix representation corresponds to a particular model of what constitutes a smooth hypergraph signal.
- Matrix-based representations: Matrix representations enable direct use of graph algorithms but may lose higher-order information, while star expansions preserve incidence information at the cost of producing bipartite graphs.Clique and line-graph constructions encode pairwise relations, whereas star expansions retain vertex–hyperedge incidence structure.
- Tensor-based representations: Tensor representations can be richer but have more complicated computations, lack algorithmic guarantees, and remain less developed for non-homogeneous hypergraphs.Determining tensor rank is NP-hard, and how to best use tensor representations for non-homogeneous hypergraphs remains open.
6. Signal processing and learning on hypergraphs
Signal processing on hypergraphs can use matrix or tensor representations for Fourier analysis, embeddings, denoising, interpolation, and neural networks. These choices produce different notions of smoothness and involve trade-offs in flexibility, computational structure, and methodological maturity.
- Fourier analysis: Matrix-based hypergraph shift operators support Fourier modes from eigenvectors, allowing many graph signal processing tools to transfer directly.Undirected hypergraphs can still yield asymmetric matrices when hyperedge weightings are included, requiring tools from directed-graph signal processing.
- Fourier analysis: Tensor-based Fourier analysis uses an orthogonal CP decomposition to derive Fourier modes and complete a basis when the tensor rank is smaller than the signal dimension.The tensor Fourier basis is formed from decomposition vectors and additional vectors satisfying specified completion conditions.
- Node embeddings: Matrix and tensor Fourier modes can both support hypergraph node embeddings, although tensor-based embeddings have received limited attention.Matrix-based embeddings follow the graph procedure, while tensor-based embeddings proceed after tensor decomposition produces the Fourier modes.
- Denoising: Hypergraph denoising estimates a smooth signal by balancing fidelity to noisy observations with a structure-dependent regularizer.Regularizers may use clique-expansion Laplacians, tensor-based total variation, or Lovász-extension formulations that induce different notions of smoothness.
- Signal interpolation: Hypergraph interpolation constrains estimates to match observed nodes while using a regularizer, whose optimal form depends on the application and may be matrix- or tensor-based.Tensor-based regularizers can provide smoothing and interpolation strategies unavailable through matrix-based approaches, while submodular Laplacian systems generalize graph Laplacian systems.
- Neural networks: Hypergraph neural networks remain a nascent research area, with one approach applying graph neural networks to shifts derived from hypergraph expansions.Clique expansion can create dense graphs because each hyperedge becomes a number of edges quadratic in its size.
7. Discussion
Higher-order signal processing addresses the inability of graphs to represent polyadic dependencies, but its theoretical foundations remain less developed than those of graph signal processing. The tutorial identifies construction choices and broader hypergraph modeling capabilities as central directions for future work.
- Discussion: Graph signal processing is successful across applications, but pairwise graph representations cannot capture higher-order dependencies in many complex systems.Simplicial complexes and hypergraphs are presented as frameworks for addressing this modeling limitation.
- Discussion: The theory of signal processing on higher-order networks is less developed than graph signal processing, motivating work on shift operators, frequency representations, smoothing, interpolation, and nonlinear architectures.The tutorial frames these topics as core components of an emerging research area.
- Future directions: Constructing simplicial complexes from data requires further study because face selection directly affects the frequency representation of signals.The discussion also identifies appropriate weighting as an open issue.
- Future directions: Hypergraph construction is key to performance, especially for large-scale real-world datasets, and more general models are needed for heterogeneous vertices and relation levels.The paper also lists spectral theory, learning, diffusion, and tensor-based operations among future research directions.