Structural properties of spatially embedded networks

• Published in: Europhysics letters Vol. 82; no. 4; pp. 48005 - p1-48005-p5
• Main Authors: Kosmidis, K, Havlin, S, Bunde, A
• Format: Journal Article
• Language: English
• Published: IOP Publishing 01.05.2008; EDP Sciences
• Subjects: 05.10.Ln; 89.75.-k; 89.75.Da
• ISSN: 0295-5075; 1286-4854
• DOI: 10.1209/0295-5075/82/48005

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$^{-\delta}$. 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 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” $l\sim$ 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 $l\sim$ 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.