Structural and functional properties of spatially embedded scale-free networks.

Scale-free networks have been studied mostly as non-spatially embedded systems. However, in many realistic cases, they are spatially embedded and these constraints should be considered. Here, we study the structural and functional properties of a model of scale-free (SF) spatially embedded networks. In our model, both the degree and the length of links follow power law distributions as found in many real networks. We show that not all SF networks can be embedded in space and that the largest degree of a node in the network is usually smaller than in nonembedded SF networks. Moreover, the spatial constraints (each node has only few neighboring nodes) introduce degree-degree anticorrelations (disassortativity) since two high degree nodes cannot stay close in space. We also find significant effects of space embedding on the hopping distances (chemical distance) and the vulnerability of the networks.