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

TL;DR

This tutorial introduces signal processing techniques on higher-order networks like simplicial complexes and hypergraphs, covering Fourier analysis, denoising, interpolation, embeddings, and neural networks, with emphasis on the Hodge Laplacian and tensor representations.

Contribution

It provides a comprehensive didactic overview of processing signals on higher-order network structures, highlighting mathematical tools and discussing trade-offs and future research directions.

Findings

Use of Hodge Laplacian for simplicial complex signal processing

Comparison of matrix and tensor representations for hypergraphs

Identification of limitations and future research avenues in higher-order network processing

Abstract

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.