Issue
EPL
Volume 82, Number 4, May 2008
Article Number 48005
Number of page(s) 5
Section Interdisciplinary Physics and Related Areas of Science and Technology
DOI http://dx.doi.org/10.1209/0295-5075/82/48005
Published online 13 May 2008
EPL, 82 (2008) 48005
DOI: 10.1209/0295-5075/82/48005

Structural properties of spatially embedded networks

K. 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

Kosmas.Kosmidis@theo.physik.uni-giessen.de

received 23 November 2007; accepted in final form 3 April 2008; published May 2008
published online 13 May 2008

Abstract
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 $\delta$ 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 $l\sim$ 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.

PACS
89.75.-k - Complex systems.
89.75.Da - Systems obeying scaling laws.
05.10.Ln - Monte Carlo methods.

© EPLA 2008