Structural properties of spatially embedded networks

We study the effects of spatial constraints on the structural properties of networks embedded in one- or two-dimensional space. When nodes are embedded in space, they have a well-defined Euclidean distance r between any pair. We assume that nodes at distance r have a link with probability p(r)∼r ^-δ. We study the mean topological distance l and the clustering coefficient C of these networks and find that they both exhibit phase transitions for some critical value of the control parameter δ depending on the dimensionality d of the embedding space. We have identified three regimes. When δ<d, the networks are not affected at all by the spatial constraints. They are "small-worlds"l∼log N with zero clustering at the thermodynamic limit. In the intermediate regime d<δ<2d, the networks are affected by the space and the distance increases and becomes a power of log N, and have non-zero clustering. When δ>2d the networks are "large" worlds l∼N^1/d with high clustering. Our results indicate that spatial constrains have a significant impact on the network properties, a fact that should be taken into account when modeling complex networks.