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 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. Copyright c EPLA, 2008

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