Effective-Resistance Preserving Spectral Reduction of Graphs

Abstract

This paper proposes a scalable algorithmic framework for effectiveresistance preserving spectral reduction of large undirected graphs. The proposed method allows computing much smaller graphs while preserving the key spectral (structural) properties of the original graph. Our framework is built upon the following three key components: a spectrum-preserving node aggregation and reduction scheme, a spectral graph sparsification framework with iterative edge weight scaling, as well as effective-resistance preserving postscaling and iterative solution refinement schemes. By leveraging recent similarity-aware spectral sparsification method and graphtheoretic algebraic multigrid (AMG) Laplacian solver, a novel constrained stochastic gradient descent (SGD) optimization approach has been proposed for achieving truly scalable performance (nearlylinear complexity) for spectral graph reduction. We show that the resultant spectrally-reduced graphs can robustly preserve the first few nontrivial eigenvalues and eigenvectors of the original graph Laplacian and thus allow for developing highly-scalable spectral graph partitioning and circuit simulation algorithms.

GRASS: Graph Spectral Sparsification Leveraging Scalable Spectral Perturbation Analysis

“…In the rest of this paper, we assume that G = (V, E, w) is a weighted, undirected and connected graph, whereas P = (V, E s , w s ) is its sparsifier. To simplify the our analysis, we assume the edge weights in the sparsifier remain the same as the original ones, though the latest iterative edge re-scaling schemes [36] can be applied to further improve the approximation. The descending eigenvalues of L + P L G are denoted by λ max = λ 1 ≥ λ 2 ≥ · · · ≥ λ n ≥ 1, where L + P denotes the Moore-Penrose pseudoinverse of L P .…”

Sf-Grass

“…Examples include scientific computing and numerical optimization [8,13,26], graph partitioning and data clustering [15,22], machine learning and data mining [6,14], as well as integrated circuit modeling, simulation and verifications [11,29,30]. In particular, latest theoretical breakthroughs in spectral graph theory have led to the development of nearly-linear time spectral graph sparsification [8,9,16,25] and coarsening algorithms [18,19,31,33]. These techniques can efficiently produce much smaller graphs that well preserve the key spectral properties of the original graph (e.g., the first few eigenvalues and eigenvectors of the graph Laplacian), which in turn has led to much faster algorithms for solving partial differential equations (PDEs) and linear systems of equations [21,25,32], spectral clustering and graph partitioning [9,15,22,31], and dimensionality reduction and data visualization [33].…”

“…This paper for the first time introduces a solver-free spectral graph sparsification framework (SF-GRASS) by leveraging emerging spectral graph coarsening [31] and graph signal processing techniques [23]. Our approach first coarsens the original graph into increasingly smaller graphs while preserving the key graph spectral properties.…”

“…Our approach first coarsens the original graph into increasingly smaller graphs while preserving the key graph spectral properties. Since spectral graph coarsening [31] can be considered as a cascade of low-pass graph filters with decreasing bandwidths, spectrally-critical edges for different ranges of eigenvalues can be effectively identified on coarse-level graphs in a stratified manner. For example, considering a coarsest graph that has only a few (e.g.…”

“…(1) For the first time, we present a solver-free spectral graph sparsification framework (SF-GRASS) by leveraging emerging spectral graph coarsening [31] and graph signal processing techniques [23]. It can be implemented using simple sparse-matrix-vector multiplications and thus completely addresses the computational challenges in prior methods that strongly reply on efficient graph Laplacian solvers.…”