The Laplacian Spectrum of Large Graphs Sampled From Graphons

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