Towards Scalable Spectral Sparsification of Directed Graphs

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 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 linearsized 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.

Mass-conserving diffusion-based dynamics on graphs

“…If G is directed, then there are a number of different approaches to defining the Laplacian on a directed graph. For example, in [32, p. 6] and [39], the unnormalised Laplacian is defined by u = D − A where A is the (directed) adjacency matrix and D is the diagonal matrix of outdegrees. An alternative approach, found in [40], is as follows: given a directed graph G = (V , E), define vertex sets H, A ⊆ V where H are the vertices with positive out-degrees d out i and A are the vertices with positive in-degrees d in i (note that H ∩ A need not be empty).…”

Sparsification of Directed Graphs via Cut Balance

“…However, their directed spectral sparsifiers only work for Eulerian graphs, i.e., for β = 1. Zhang et al [49] proposed a notion of spectral sparsification that works for all directed graphs, but their definition does not preserve cut values. More generally, there have been attempts at bridging the divide between directed and undirected graphs for other problems.…”

High-Performance Spectral Methods for Computer-Aided Design of Integrated Circuits

“…For example, spectral methods can potentially lead to much faster algorithms for solving sparse matrices [53,116], numerical optimization [12], data mining [78,103], graph analytics [39,49], machine learning [18,19], as well as very-large-scale integration (VLSI) computeraided design (CAD) [27,28,110,112,113,116]. To this end, spectral sparsification of graphs has been extensively studied in the past decade [5,56,93,94,111] to generate almost-linear-sized 1 subgraphs or sparsifiers that can robustly preserve the spectrum 1 The number of vertices (nodes) is similar to the number of edges.…”

“…of the original graph Laplacian. The sparsified graphs retain the same set of vertices but much fewer edges, which can be regarded as ultra-sparse graph proxies and have been leveraged for developing a series of nearly-linear-time numerical and graph algorithms [11,32,92,93,109]. Another way of simplifying graphs is to directly reduce the size of the graphs, which is widely used in many areas, including graph partitioning [43], machine learning [18] and multigrid solvers [51,64].…”