Graphon Signal Processing, Luana Ruiz

Graphons are infinite dimensional objects that represent the limit of convergent sequences of discrete graphs. This talk discusses a theory of Graphon Signal Processing centered on the notions of graphon Fourier transform and linear shift invariant graphon filters. These two objects are graphon counterparts of graph Fourier transforms and graph filters. It is shown that in convergent sequences of graphs and associated graph signals: (i) The graph Fourier transform converges to the graphon Fourier transform when considering graphon bandlimited signals. (ii) The spectral and vertex responses of graph filters converge to the spectral and vertex responses of graphon filters with the same coefficients. These theorems imply that for graphs that belong to certain families —in the sense that they are part of sequences that converge to a certain graphon— graph Fourier analysis and graph filter design have well defined limits. In turn, these facts extend applicability of graph signal processing to graphs with large number of nodes —because we can transfer designs from limit graphons to finite graphs— and dynamic graphs —because we can transfer designs to different graphs drawn from the same graphon.