Lecture 9: Graphon Signal Processing (10/25 – 10/29)
In this lecture we discuss graphon signal processing. A graphon is a bounded function defined on the unit square that can be conceived as the limit of a sequence of graphs whose number of nodes and edges grows up to infinity. This framework provides a powerful set of tools and insights that facilitate the understanding of structures like GNNs when the number of nodes in the graph layers is large. By means of graphons we can translate graph signal processing into harmonic analysis on the unit interval allowing us to exploit consolidated tools from functional analysis. This provides indeed a beautiful connection of two notions of Fourier decompositions apparently disconnected.
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• The graphon Fourier transform converges to the graphon Fourier transform (Proof)
• Frequency representation of graphon filters. (Proof)
Video 9.1 – Definitions and Examples
In this part of the lecture we define and motivate graphons as a way of understanding the limit behavior of graphs with a large number of nodes. We show how graphons capture structural similarities in a family of graphs to both generate random graphs and capture the limit behavior of graph sequences. This is illustrated using examples on uniform and stochastic block model graphons.
• Covers Slides 1-10 in the handout.
Video 9.2 – Convergence of Graph Sequences
In this lecture, we show that the graphon is the limit object of a convergent graph sequence. We start by introducing three concepts: motifs, homomorphisms and homomorphism densities, and later state formally the graph convergence in the homomorphism density. We provide examples illustrating convergent graph sequences with their associated graphons. The induced graphons are defined, which constructs a graphon for each undirected graph.
• Covers Slides 11-20 in the handout.
Video 9.3 – Graphon signals
In this part of the lecture, we define graphon signals and the graphon shift operator (WSO). We present graphon signals as both a generating model for graph signals and the limit of a convergence sequence of graph signals. We build the theoretical grounds of this twofold phenomena and we provide the intuition of why this is happenings. Regarding the graphon shift operator, we define it and we build the connection with its graph counterpart. We conclude the lecture by defining the graphon diffusion as a recursive sequence.
• Covers Slides 21-26 in the handout.
Video 9.4 – Graphon Fourier Transform (WFT)
Our next goal is to generalize the notions and concepts of graph signal processing to graphons, starting with the definition of a Fourier transform for graphon signals. The graphon Fourier transform (WFT) is a change of basis from the original domain to the basis defined by eigenfunctions of the graphon shift operator. We first discuss the eigenvalues and eigenfunctions of a graphon shift operator, and then show that the eigenfunctions form a complete orthonormal basis of L^2([0,1]). That will allow us to define the graphon Fourier transform (WFT) and its inverse, the iWFT.
• Covers Slides 27-36 in the handout.
Video 9.5 – The GFT converges to the WFT
In this lecture we discuss the convergence of the GFT for sequences of graph signals whose limit is a graphon signal. Exploiting the representation of graphs as induced graphons, we show that when the equivalent graphon signals are bandlimited their WFT converges to the WFT to the limit signal. In showing this we discuss the challenges that we face when considering a convergence analysis with graphons in the spectral domain, in particular the accumulation of the eigenvalues of the graphon at 0 and how this affect the convergence of the eigenspaces. The proof of the convergence of the GFT to the WFT can be found here.
• Covers Slides 37-48 in the handout.
Video 9.6 – Graphon Filters
In this part of the lecture we define graphon filters. We further study the frequency response of graphon filters and show its independence of graphon. It also turns out that the graphon filters are point wise in the graphon Fourier transform domain. This is also the exact same definition of the frequency response of a graph filter with the same coefficients. The proof for the frequency representation of graphon filters can be found here.
• Covers Slides 49-54 in the handout.