Unified Fourier transform on graphs sampled from stochastic block models
This work provides a theoretical framework for Fourier analysis on SBM graphs, benefiting graph signal processing researchers by offering a computationally simpler basis and sensitivity analysis.
The paper extends graphon-based Fourier transforms to graphs sampled from stochastic block models (SBMs), deriving a Fourier basis from block sizes and probability matrix, with perturbation bounds on sensitivity. It shows that for equal block sizes, a group-theoretic basis matches the SBM basis, and for nearly uniform sizes it approximates it, while for non-uniform sizes the group still provides partial information.
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 For highly non-uniform block sizes, the group-based Fourier basis is no longer applicable, though, as we show, the underlying group still provides partial information about the SBM Fourier basis.