Estimating Current-Flow Closeness Centrality with a Multigrid Laplacian Solver

TL;DR

This paper introduces fast algorithms and an implementation for estimating current-flow closeness centrality in large networks using multigrid Laplacian solvers, enabling practical analysis of massive graphs.

Contribution

It presents two novel algorithms for accelerated current-flow closeness computation and integrates a multigrid solver into NetworKit for scalable network analysis.

Findings

Current-flow closeness better discriminates nodes than shortest-path closeness.

Algorithms enable estimation on networks with tens of millions of nodes within seconds.

Current-flow closeness is more resistant to noise than traditional measures.

Abstract

Matrices associated with graphs, such as the Laplacian, lead to numerous interesting graph problems expressed as linear systems. One field where Laplacian linear systems play a role is network analysis, e. g. for certain centrality measures that indicate if a node (or an edge) is important in the network. One such centrality measure is current-flow closeness. To allow network analysis workflows to profit from a fast Laplacian solver, we provide an implementation of the LAMG multigrid solver in the NetworKit package, facilitating the computation of current-flow closeness values or related quantities. Our main contribution consists of two algorithms that accelerate the current-flow computation for one node or a reasonably small node subset significantly. One sampling-based algorithm provides an unbiased estimation of the related electrical farness, the other one is based on the Johnson-Lindenstrauss transform. Our inexact algorithms lead to very accurate results in practice. Thanks to them one is now able to compute an estimation of current-flow closeness of one node on networks with tens of millions of nodes and edges within seconds or a few minutes. From a network analytical point of view, our experiments indicate that current-flow closeness can discriminate among different nodes significantly better than traditional shortest-path closeness and is also considerably more resistant to noise -- we thus show that two known drawbacks of shortest-path closeness are alleviated by the current-flow variant.