Spatially embedded random networks
Abstract
Many realworld networks analyzed in modern network theory have a natural spatial element; e.g., the Internet, social networks, neural networks, etc. Yet, aside from a comparatively small number of somewhat specialized and domainspecific studies, the spatial element is mostly ignored and, in particular, its relation to network structure disregarded. In this paper we introduce a model framework to analyze the mediation of network structure by spatial embedding; specifically, we model connectivity as dependent on the distance between network nodes. Our spatially embedded random networks construction is not primarily intended as an accurate model of any specific class of realworld networks, but rather to gain intuition for the effects of spatial embedding on network structure; nevertheless we are able to demonstrate, in a quite general setting, some constraints of spatial embedding on connectivity such as the effects of spatial symmetry, conditions for scale free degree distributions and the existence of smallworld spatial networks. We also derive some standard structural statistics for spatially embedded networks and illustrate the application of our model framework with concrete examples.
 Publication:

Physical Review E
 Pub Date:
 November 2007
 DOI:
 10.1103/PhysRevE.76.056115
 Bibcode:
 2007PhRvE..76e6115B
 Keywords:

 89.75.Hc;
 05.10.Ln;
 64.60.Ak;
 89.75.Da;
 Networks and genealogical trees;
 Monte Carlo methods;
 Renormalizationgroup fractal and percolation studies of phase transitions;
 Systems obeying scaling laws