Crossover from scalefree to spatial networks
Abstract
In many networks such as transportation or communication networks, distance is certainly a relevant parameter. In addition, realworld examples suggest that when longrange links are existing, they usually connect to hubs—the wellconnected nodes. We analyze a simple model which combines both these ingredients—preferential attachment and distance selection characterized by a typical finite "interaction range". We study the crossover from the scalefree to the "spatial" network as the interaction range decreases and we propose scaling forms for different quantities describing the network. In particular, when the distance effect is important i) the connectivity distribution has a cutoff depending on the node density, ii) the clustering coefficient is very high, and iii) we observe a positive maximum in the degree correlation (assortativity) whose numerical value is in agreement with empirical measurements. Finally, we show that if the total length is fixed, the optimal network which minimizes both the total length and the diameter lies in between the scalefree and spatial networks. This phenomenon could play an important role in the formation of networks and could be an explanation for the high clustering and the positive assortativity which are nontrivial features observed in many realworld examples.
 Publication:

EPL (Europhysics Letters)
 Pub Date:
 September 2003
 DOI:
 10.1209/epl/i2003006006
 arXiv:
 arXiv:condmat/0212086
 Bibcode:
 2003EL.....63..915B
 Keywords:

 89.75.Fb;
 89.75.Hc;
 05.40.a;
 Structures and organization in complex systems;
 Networks and genealogical trees;
 Fluctuation phenomena random processes noise and Brownian motion;
 Condensed Matter  Disordered Systems and Neural Networks;
 Condensed Matter  Statistical Mechanics
 EPrint:
 4 pages, 6 figures, final version