Spectral classes of regular, random, and empirical graphs

TL;DR

This paper introduces a spectral distance based on the normalized Laplacian to classify large graphs into families with similar spectral properties, demonstrating its practical usefulness through numerical experiments.

Contribution

It defines a new spectral distance for graphs and shows its effectiveness in classifying large empirical graphs based on spectral similarity.

Findings

Spectral distance is easy to compute or estimate numerically.

Graphs within the same family have bounded spectral distance as size increases.

Numerical experiments confirm the spectral distance's practical utility.

Abstract

We define a (pseudo-)distance between graphs based on the spectrum of the normalized Laplacian, which is easy to compute or to estimate numerically. It can therefore serve as a rough classification of large empirical graphs into families that share the same asymptotic behavior of the spectrum so that the distance of two graphs from the same family is bounded by $\mathcal{O}(1/n)$ in terms of size $n$ of their vertex sets. Numerical experiments demonstrate that the spectral distance provides a practically useful measure of graph dissimilarity.