Unified Fourier bases for GSP on stochastic block model graphs

We consider a recently proposed approach to graph signal processing based on graphons. We show how the graphon-based approach to GSP applies to graphs sampled from a stochastic block model. We obtain a basis for the graphon Fourier transform on such samples directly from the link probability matrix and the block sizes of the model. This formulation allows us to bound the sensitivity of the Fourier transform to small changes in block sizes. We then focus on the case where the probability matrix corresponds to a (weighted) Cayley graph. If block sizes are equal, a nice Fourier basis can be derived from the underlying group. We explore how, in the case where block sizes are not equal, some or all nice properties of the group basis can be maintained. We complement the theoretical results with simulations.

Signal processing on large networks with group symmetries

Current methods of graph signal processing rely heavily on the specific structure of the underlying network: the shift operator and the graph Fourier transform are both derived directly from a specific graph. In many cases, the network is subject to error or natural changes over time. This motivated a new perspective on GSP, where the signal processing framework is developed for an entire class of graphs with similar structures. This approach can be formalized via the theory of graph limits, where graphs are considered as random samples from a distribution represented by a graphon. When the network under consideration has underlying symmetries, they may be modeled as samples from Cayley graphons. In Cayley graphons, vertices are sampled from a group, and the link probability between two vertices is determined by a function of the two corresponding group elements. Infinite groups such as the 1-dimensional torus can be used to model networks with an underlying spatial reality. Cayley graphons on finite groups give rise to a Stochastic Block Model, where the link probabilities between blocks form a (edge-weighted) Cayley graph. This manuscript summarizes some work on graph signal processing on large networks, in particular samples of Cayley graphons.