A Unified Spectral Sparsification Framework for Directed Graphs

ying zhang, Zhiqiang Zhao, Zhuo Feng

Keywords: Spectral Graph Theory, Spectral Sparsification, Directed Graphs, Laplacian Solver, PageRank Vectors

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. For the first time, 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.

One-sentence Summary: A unified directed graphs spectral sparsification approach that is more general than prior theoretical results

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