A Local Clustering Algorithm for Massive Graphs and its Application to Nearly-Linear Time Graph Partitioning

TL;DR

This paper introduces a local clustering algorithm for massive graphs that operates in nearly linear time, enabling efficient graph partitioning, spectral sparsifier construction, and solving linear systems in large-scale graphs.

Contribution

The paper presents a novel local clustering algorithm with nearly linear time complexity and demonstrates its application to graph partitioning and spectral sparsification.

Findings

Efficient local clustering with nearly linear time complexity.

Approximate sparsest cut with nearly optimal balance in nearly linear time.

Construction of spectral sparsifiers and solutions to linear systems in large graphs.

Abstract

We study the design of local algorithms for massive graphs. A local algorithm is one that finds a solution containing or near a given vertex without looking at the whole graph. We present a local clustering algorithm. Our algorithm finds a good cluster--a subset of vertices whose internal connections are significantly richer than its external connections--near a given vertex. The running time of our algorithm, when it finds a non-empty local cluster, is nearly linear in the size of the cluster it outputs. Our clustering algorithm could be a useful primitive for handling massive graphs, such as social networks and web-graphs. As an application of this clustering algorithm, we present a partitioning algorithm that finds an approximate sparsest cut with nearly optimal balance. Our algorithm takes time nearly linear in the number edges of the graph. Using the partitioning algorithm of this paper, we have designed a nearly-linear time algorithm for constructing spectral sparsifiers of graphs, which we in turn use in a nearly-linear time algorithm for solving linear systems in symmetric, diagonally-dominant matrices. The linear system solver also leads to a nearly linear-time algorithm for approximating the second-smallest eigenvalue and corresponding eigenvector of the Laplacian matrix of a graph. These other results are presented in two companion papers.