Signal Processing on Higher-Order Networks: Livin' on the Edge ... and Beyond

Abstract

This tutorial paper presents 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 abstractions of higher-order networks that can incorporate polyadic relationships. We provide basic introductions to simplicial complexes and hypergraphs, making special emphasis on the concepts needed for processing signals on them. Leveraging these concepts, we discuss Fourier analysis, signal denoising, signal interpolation, node embeddings, and non-linear processing through neural networks in these two representations of polyadic relational structures. 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.

Simplicial Convolutional Neural Networks

“…For example, R N0 is the graph signal space in GSP, and R N1 is the space of edge flows. For an edge flow x 1 ∈ R N1 , the sign of its entry denotes the direction of the flow relative to a chosen reference orientation [11,17]. Hodge Laplacian and decomposition.…”

“…, K. The rows of B k are indexed by (k − 1)-simplices and the columns by k-simplices. E.g., matrix B 1 is the node-to-edge incidence matrix, and B 2 is the edge-to-triangle incidence matrix [11,12].…”

“…where ⊕ is the direct sum of vector spaces and im(•) and ker(•) are the image and kernel of a matrix. For k = 1, these subspaces carry the following interpretations [7,11].…”

Finite Impulse Response Filters for Simplicial Complexes

“…Similarly, in a social network, we want to capture the connections between groups of friends rather than two people [8]. To deal with such settings, researchers have introduced hypergraph signal processing as one paradigm, in which we model group relationships through hyperedges [9], [10], [11]. Other works [12], [13] use the Volterra model to describe higher-order relations.…”

“…In parallel, topological signal processing (TSP) has been proposed to analyze signals defined over topological spaces, especially in the form of simplicial complexes composed of nodes, edges, and triangles, etc. [11], [14], [15]. Important examples of such signals defined on simplices include flow signals over edges, like traffic flows in a transportation network or data flows in a communication network.…”