On the Approximation of Laplacian Eigenvalues in Graph Disaggregation

TL;DR

This paper investigates how graph disaggregation affects the spectral properties of the Laplacian, providing theoretical bounds and a new preconditioning approach to improve computational efficiency for large graphs.

Contribution

It offers new theoretical results on the spectral approximation of Laplacians after disaggregation and introduces an alternate operator with interlaced eigenvalues for better preconditioning.

Findings

Spectral approximation bounds for disaggregated Laplacians

Construction of an alternate disaggregation operator with interlaced eigenvalues

Development of a uniform preconditioner for the original Laplacian

Abstract

Graph disaggregation is a technique used to address the high cost of computation for power law graphs on parallel processors. The few high-degree vertices are broken into multiple small-degree vertices, in order to allow for more efficient computation in parallel. In particular, we consider computations involving the graph Laplacian, which has significant applications, including diffusion mapping and graph partitioning, among others. We prove results regarding the spectral approximation of the Laplacian of the original graph by the Laplacian of the disaggregated graph. In addition, we construct an alternate disaggregation operator whose eigenvalues interlace those of the original Laplacian. Using this alternate operator, we construct a uniform preconditioner for the original graph Laplacian.