Structural properties of spatially embedded networks

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

Corresponding author: Kosmas.Kosmidis@theo.physik.uni-giessen.de

Received: 23 November 2007
Accepted: 3 April 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. 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” 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 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.