RWTH Aachen University

We are confronted with signals defined on the nodes of a graph in many applications.  Think for instance of a sensor network measuring temperature; or a social network, in which each person (node) has an opinion about a specific issue.  Graph signal processing (GSP) tries to device appropriate tools to process such data by generalizing classical methods from signal processing of time-series and images -- such as smoothing, filtering and interpolation -- to signals defined on graphs.  Typically, this involves leveraging the structure of the graph as encoded in the spectral properties of the graph Laplacian.

In other applications such as traffic network analysis, however, the signals of interest are naturally defined on the edges of a graph, rather than on the nodes. After a very brief recap of the central ideas of GSP, we examine why the standard tools from GSP may not be suitable for the analysis of such edge signals.  More specifically, we discuss how the underlying notion of a 'smooth signal' inherited from (the typically considered variants of) the graph Laplacian are not suitable when dealing with edge signals that encode flows.  To overcome this limitation we devise signal processing tools based on the Hodge-Laplacian and the associated discrete Hodge Theory for simplicial (and cellular) complexes.  We discuss applications of these ideas for signal smoothing, semi-supervised and active learning for edge-flows on discrete (or discretized) spaces.

Multibody interactions can reveal higher-order dynamical effects that are not captured by traditional two-body network models. We derive and analyze models for consensus dynamics on hypergraphs, where nodes interact in groups rather than in pairs. Our work reveals that multibody dynamical effects that go beyond rescaled pairwise interactions can appear only if the interaction function is nonlinear, regardless of the underlying multibody structure. As a practical application, we introduce a specific nonlinear function to model three-body consensus, which incorporates reinforcing group effects such as peer pressure. Unlike consensus processes on networks, we find that the resulting dynamics can cause shifts away from the average system state. The nature of these shifts depends on a complex interplay between the distribution of the initial states, the underlying structure, and the form of the interaction function. By considering modular hypergraphs, we discover state-dependent, asymmetric dynamics between polarized clusters where multibody interactions make one cluster dominate the other.

Building on these results, we generalise the model allowing for interactions within hyper edges of any cardinality and explore in detail the role of involvement and stubbornness on polarisation.