The Laplacian Spectrum of Large Graphs Sampled from Graphons

This paper studies the Laplacian spectrum and the average effective resistance of (large) graphs that are sampled from graphons. Broadly speaking, our main finding is that the Laplacian eigenvalues of a large dense graph can be effectively approximated by using the degree function of the corresponding graphon. More specifically, we show how to approximate the distribution of the Laplacian eigenvalues and the average effective resistance (Kirchhoff index) of the graph. For all cases, we provide explicit bounds on the approximation errors and derive the asymptotic rates at which the errors go to zero when the number of nodes goes to infinity. Our main results are proved under the conditions that the graphon is piecewise Lipschitz and bounded away from zero.

Graphon-based sensitivity analysis of SIS epidemics

In this work, we use the spectral properties of graphons to study stability and sensitivity to noise of deterministic SIS epidemics over large networks. We consider the presence of additive noise in a linearized SIS model and we derive a noise index to quantify the deviation from the disease-free state due to noise. For finite networks, we show that the index depends on the adjacency eigenvalues of its graph. We then assume that the graph is a random sample from a piecewise Lipschitz graphon with finite rank and, using the eigenvalues of the associated graphon operator, we find an approximation of the index that is tight when the network size goes to infinity. A numerical example is included to illustrate the results.