Crossover from scale-free to spatial networks

In many networks such as transportation or communication networks, distance is certainly a relevant parameter. In addition, real-world examples suggest that when long-range links are existing, they usually connect to hubs—the well-connected 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 scale-free 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 cut-off 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 scale-free 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 non-trivial features observed in many real-world examples.