A Unified Approach to Scalable Spectral Sparsification of Directed Graphs

TL;DR

This paper introduces a unified, scalable spectral sparsification method for both directed and undirected graphs, preserving spectral properties and enabling efficient large-scale graph analysis.

Contribution

It proves the existence of linear-sized spectral sparsifiers for directed graphs and develops a practical, unified approach with nearly-linear complexity.

Findings

Successfully sparsifies large directed graphs from public datasets

Preserves key eigenvalues and eigenvectors of graph Laplacians

Enhances efficiency in spectral embedding and PageRank computations

Abstract

Recent spectral graph sparsification research allows constructing nearly-linear-sized subgraphs that can well preserve the spectral (structural) properties of the original graph, such as the first few eigenvalues and eigenvectors of the graph Laplacian, leading to the development of a variety of nearly-linear time numerical and graph algorithms. However, there is not a unified approach that allows for truly-scalable spectral sparsification of both directed and undirected graphs. In this work, we prove the existence of linear-sized spectral sparsifiers for general directed graphs and introduce a practically-efficient and unified spectral graph sparsification approach that allows sparsifying real-world, large-scale directed and undirected graphs with guaranteed preservation of the original graph spectra. By exploiting a highly-scalable (nearly-linear complexity) spectral matrix perturbation analysis framework for constructing nearly-linear sized (directed) subgraphs, it enables us to well preserve the key eigenvalues and eigenvectors of the original (directed) graph Laplacians. The proposed method has been validated using various kinds of directed graphs obtained from public domain sparse matrix collections, showing promising results for solving directed graph Laplacians, spectral embedding, and partitioning of general directed graphs, as well as approximately computing (personalized) PageRank vectors.