Structural properties of spatially embedded networksK. Kosmidis1, S. Havlin2 and A. Bunde1
1 Institut für Theoretische Physik III, Justus-Liebig-Universität Giessen - 35392 Giessen,Germany, EU
2 Minerva Center and Department of Physics, Bar-Ilan University - Ramat-Gan 52900, Israel
received 23 November 2007; accepted in final form 3 April 2008; published May 2008
published online 13 May 2008
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 . 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 depending on the dimensionality d of the embedding space. We have identified three regimes. When < d, the networks are not affected at all by the spatial constraints. They are "small-worlds" log N with zero clustering at the thermodynamic limit. In the intermediate regime d < < 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 > 2d the networks are "large" worlds N1/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.
89.75.-k - Complex systems.
89.75.Da - Systems obeying scaling laws.
05.10.Ln - Monte Carlo methods.
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