Towards Scalable Spectral Sparsification of Directed Graphs
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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 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, this paper proves the existence of linear-sized spectral sparsifiers for general directed graphs, and introduces a practically-efficient yet 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 to well preserve the key eigenvalues and eigenvectors of the original (directed) graph Laplacians. Compared with prior works that are limited to only strongly-connected directed graphs, the proposed approach is more general and thus will allow for truly-scalable spectral sparsification of a much wider range of real-world complex graphs. The proposed method has been validated using various kinds of directed graphs obtained from public domain sparse matrix collections, showing promising spectral sparsification and partitioning results for general directed graphs.
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