  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.
