Similarity-Aware Spectral Sparsification by Edge Filtering

TL;DR

This paper introduces a similarity-aware spectral graph sparsification method that uses spectral embedding and filtering to create ultra-sparse graph proxies with guaranteed spectral similarity, improving efficiency for large-scale graph applications.

Contribution

It proposes a novel spectral sparsification framework leveraging spectral embedding and filtering, with an iterative densification scheme for better approximation quality.

Findings

Effective spectral sparsifiers with guaranteed similarity levels

Validated on graphs from diverse domains including VLSI, finite element, and social networks

Improved efficiency in large-scale graph processing

Abstract

In recent years, spectral graph sparsification techniques that can compute ultra-sparse graph proxies have been extensively studied for accelerating various numerical and graph-related applications. Prior nearly-linear-time spectral sparsification methods first extract low-stretch spanning tree from the original graph to form the backbone of the sparsifier, and then recover small portions of spectrally-critical off-tree edges to the spanning tree to significantly improve the approximation quality. However, it is not clear how many off-tree edges should be recovered for achieving a desired spectral similarity level within the sparsifier. Motivated by recent graph signal processing techniques, this paper proposes a similarity-aware spectral graph sparsification framework that leverages efficient spectral off-tree edge embedding and filtering schemes to construct spectral sparsifiers with guaranteed spectral similarity (relative condition number) level. An iterative graph densification scheme is introduced to facilitate efficient and effective filtering of off-tree edges for highly ill-conditioned problems. The proposed method has been validated using various kinds of graphs obtained from public domain sparse matrix collections relevant to VLSI CAD, finite element analysis, as well as social and data networks frequently studied in many machine learning and data mining applications.