Signal Processing on Kernel-Based Random Graphs

We present the theory of sequences of random graphs and their convergence to limit objects. Sequences of random dense graphs are shown to converge to their limit objects in both their structural properties and their spectra. The limit objects are bounded symmetric functions on [0, 1](2). The kernel functions define an equivalence class and thus identify collections of large random graphs who are spectrally and structurally equivalent. As the spectrum of the graph shift operator defines the graph Fourier transform (GFT), the behavior of the spectrum of the underlying graph has a great impact on the design and implementation of graph signal processing operators such as filters. The spectra of several graph limits are derived analytically and verified with numerical examples.