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Title: The Fourier transform on Convergent Graph Sequences

Abstract:
A graph signal is a function on the vertices of a graph. Examples of graph signals are: viral load of patients connected through a contact network, or fMRI signals on brain regions linked by functional or physical connections. The graph Fourier transform is a projection of the signal onto the eigenbasis of the adjacency matrix the graph. The matrix is referred to as the shift operator and represents the time evolution of a signal over a graph.

Sequences of graphs that have similar structure, in terms of homomorphism densities, converge to a {\sl graphon}. A graphon is a symmetric, measurable function on $[0,1]^2$ which can be interpreted as a random graph model. Graphs in a graph sequence converging to a graphon have structure typical of samples from this model. I will present a common framework, derived from the graphon, to define the Fourier transform of such sampled graphs. We recently showed that this approach has the desired convergence properties. I will also discuss the special case of graphs sampled from a {\sl Cayley graphon}. Such graphs are stochastic versions of Cayley graphs. The representations of the underlying group allow us to define a basis for the graphon Fourier transform.

This is joint work with Mahya Ghandehari and Nauzer Kalyaniwalla.

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