Graphon Signal Processing
Abstract
Graphons are infinitedimensional objects that represent the limit of convergent sequences of graphs as their number of nodes goes to infinity. This paper derives a theory of graphon signal processing centered on the notions of graphon Fourier transform and linear shift invariant graphon filters, the graphon counterparts of the graph Fourier transform and graph filters. It is shown that for convergent sequences of graphs and associated graph signals: (i) the graph Fourier transform converges to the graphon Fourier transform when the graphon signal is bandlimited; (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, i.e., that 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  since signal processing pipelines designed for limit graphons can be applied to finite graphs  and to dynamic graphs  since we can relate the result of SP pipelines designed for different graphs from the same convergent graph sequence.
 Publication:

IEEE Transactions on Signal Processing
 Pub Date:
 2021
 DOI:
 10.1109/TSP.2021.3106857
 arXiv:
 arXiv:2003.05030
 Bibcode:
 2021ITSP...69.4961R
 Keywords:

 Electrical Engineering and Systems Science  Signal Processing
 EPrint:
 Accepted for publication in IEEE TSP