A Cascadic Multigrid Algorithm for Computing the Fiedler Vector of Graph Laplacians

TL;DR

This paper introduces a cascadic multigrid algorithm for efficiently computing the Fiedler vector of graph Laplacians, with theoretical analysis and numerical validation demonstrating its effectiveness.

Contribution

It presents a novel algebraic cascadic multigrid method specifically designed for fast Fiedler vector computation, including convergence analysis and practical performance results.

Findings

The algorithm achieves optimal computational efficiency.

Numerical tests confirm the method's effectiveness on real-world graphs.

Theoretical analysis shows uniform convergence under certain conditions.

Abstract

In this paper, we develop a cascadic multigrid algorithm for fast computation of the Fiedler vector of a graph Laplacian, namely, the eigenvector corresponding to the second smallest eigenvalue. This vector has been found to have applications in fields such as graph partitioning and graph drawing. The algorithm is a purely algebraic approach based on a heavy edge coarsening scheme and pointwise smoothing for refinement. To gain theoretical insight, we also consider the related cascadic multigrid method in the geometric setting for elliptic eigenvalue problems and show its uniform convergence under certain assumptions. Numerical tests are presented for computing the Fiedler vector of several practical graphs, and numerical results show the efficiency and optimality of our proposed cascadic multigrid algorithm.