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 $p(r) \sim r^{- \delta}$. 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 $\delta$ depending on the dimensionality $d$ of the embedding space. We have identified three regimes. When $\delta <d$, the networks are not affected at all by the spatial constraints. They are ``small-worlds'' $l\sim \log N$ with zero clustering at the thermodynamic limit. In the intermediate regime $d<\delta<2d$, the networks are affected by the space and the distance increases and becomes a power of $\log N$, and have non-zero clustering. When $\delta>2d$ the networks are ``large'' worlds $l \sim N^{1/d}$ 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.