Extracting spatial information from networks with low-order eigenvectors

TL;DR

This paper demonstrates that low-order eigenvectors of network Laplacians can reveal meaningful spatial regions and boundaries in networks like migration and communication, even with incomplete data.

Contribution

It shows that low-order eigenvectors localize well and correspond to geographical regions, linking spectral graph theory with spatial inference in networks.

Findings

Low-order eigenvectors localize and reveal geographical regions.

Eigenvectors correspond to natural spatial subdivisions.

Numerical evidence suggests eigenvectors indicate local network cuts.

Abstract

We consider the problem of inferring meaningful spatial information in networks from incomplete information on the connection intensity between the nodes of the network. We consider two spatially distributed networks: a population migration flow network within the US, and a network of mobile phone calls between cities in Belgium. For both networks we use the eigenvectors of the Laplacian matrix constructed from the link intensities to obtain informative visualizations and capture natural geographical subdivisions. We observe that some low order eigenvectors localize very well and seem to reveal small geographically cohesive regions that match remarkably well with political and administrative boundaries. We discuss possible explanations for this observation by describing diffusion maps and localized eigenfunctions. In addition, we discuss a possible connection with the weighted graph cut problem, and provide numerical evidence supporting the idea that lower order eigenvectors point out local cuts in the network. However, we do not provide a formal and rigorous justification for our observations.