Unified Fourier transform on graphs sampled from stochastic block models

Recently, an approach to graph signal processing based on graphons was proposed. Here we show how such a graphon-driven approach to the Fourier transform can be used on graphs sampled from a stochastic block model (SBM). In particular, we show how a Fourier basis can be easily calculated from the block sizes and the block probability matrix. Using perturbation theory, we derive bounds on the sensitivity of the basis with respect to variations in the block sizes. We then consider SBMs constructed from weighted Cayley graphs. When block sizes are equal, a nice Fourier basis can be derived from the representation theory of the underlying group. When block sizes are nearly uniform, we demonstrate that this Fourier basis closely approximates the SBM Fourier basis. For highly non-uniform block sizes, the group-based Fourier basis is no longer applicable, though, as we show, the underlying group still provides partial information about the SBM Fourier basis.