Structural Analysis of Laplacian Spectral Properties of Large-Scale Networks

TL;DR

This paper explores how local structural features of large networks influence the spectral properties of their Laplacian matrices, providing new formulas and optimization methods to bound spectral characteristics.

Contribution

It introduces spectral moment expressions based on local features and semidefinite programs for bounding spectral radius and gap from spectral moments.

Findings

Spectral moments are strongly constrained by local features.

Spectral radius can be estimated from local structural measurements.

Spectral gap estimation requires more than local features.

Abstract

Using methods from algebraic graph theory and convex optimization, we study the relationship between local structural features of a network and spectral properties of its Laplacian matrix. In particular, we derive expressions for the so-called spectral moments of the Laplacian matrix of a network in terms of a collection of local structural measurements. Furthermore, we propose a series of semidefinite programs to compute bounds on the spectral radius and the spectral gap of the Laplacian matrix from a truncated sequence of Laplacian spectral moments. Our analysis shows that the Laplacian spectral moments and spectral radius are strongly constrained by local structural features of the network. On the other hand, we illustrate how local structural features are usually not enough to estimate the Laplacian spectral gap.