  Higher-order networks have so far been considered primarily in the context of
studying the structure of complex systems, i.e., the higher-order or multi-way
relations connecting the constituent entities. More recently, a number of
studies have considered dynamical processes that explicitly account for such
higher-order dependencies, e.g., in the context of epidemic spreading processes
or opinion formation. In this chapter, we focus on a closely related, but
distinct third perspective: how can we use higher-order relationships to
process signals and data supported on higher-order network structures. In
particular, we survey how ideas from signal processing of data supported on
regular domains, such as time series or images, can be extended to graphs and
simplicial complexes. We discuss Fourier analysis, signal denoising, signal
interpolation, and nonlinear processing through neural networks based on
simplicial complexes. Key to our developments is 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.
