Spatial effects in real networks: measures, null models, and   applications
  Spatially embedded networks are shaped by a combination of purely topological (space-independent) and space-dependent formation rules. While it is quite easy to artificially generate networks where the relative importance of these two factors can be varied arbitrarily, it is much more difficult to disentangle these two architectural effects in real networks. Here we propose a solution to the problem by introducing global and local measures of spatial effects that, through a comparison with adequate null models, effectively filter out the spurious contribution of non-spatial constraints. Our filtering allows us to consistently compare different embedded networks or different historical snapshots of the same network. As a challenging application we analyse the World Trade Web, whose topology is expected to depend on geographic distances but is also strongly determined by non-spatial constraints (degree sequence or GDP). Remarkably, we are able to detect weak but significant spatial effects both locally and globally in the network, showing that our method succeeds in retrieving spatial information even when non-spatial factors dominate. We finally relate our results to the economic literature on gravity models and trade globalization. 
Franco Ruzzenenti and Francesco Picciolo and Riccardo Basosi and Diego   Garlaschelli
physics.soc-ph cs.SI