  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.

Data Structures and Algorithms