Effective-Resistance Preserving Spectral Reduction of Graphs

This paper proposes a scalable algorithmic framework for effective-resistance 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.