Graphon Filters: Signal Processing in Very Large Graphs

Graph filters are at the core of network information processing architectures, with applications in machine learning and distributed collaborative systems. Yet, designing filters for very large graphs is challenging because filter design techniques do not always scale with graph size. We overcome this by using graphons, which are infinite-dimensional representations of graphs that are at once random graph models and limit objects of sequences of graphs. Explicitly, we define graphon filters and leverage convergence properties of graph sequences and spectral properties of both graphs and graphons to show that graph filters converge to graphon filters. Filters designed on a graphon can therefore be applied to finite graphs sampled from it with guaranteed convergence properties. We illustrate our findings in two experiments, which corroborate filter response convergence and illustrate transferability of graph filters even in graphs that are not related to one another through a graphon, but that are built from the same type of data.