Scale-free networks embedded in fractal space

Details Scale-free networks embedded in fractal space Scale-free networks embedded in fractal space

2010-09-03

arxiv.org/abs/1009.0580v2

Physics

Condensed Matter

Statistical Mechanics

11 pages, 10 figures

Scientific paper

The impact of inhomogeneous arrangement of nodes in space on network organization cannot be neglected in most of real-world scale-free networks. Here, we wish to suggest a model for a geographical network with nodes embedded in a fractal space in which we can tune the network heterogeneity by varying the strength of the spatial embedding. When the nodes in such networks have power-law distributed intrinsic weights, the networks are scale-free with the degree distribution exponent decreasing with increasing fractal dimension if the spatial embedding is strong enough, while the weakly embedded networks are still scale-free but the degree exponent is equal to $\gamma=2$ regardless of the fractal dimension. We show that this phenomenon is related to the transition from a non-compact to compact phase of the network and that this transition is related to the divergence of the edge length fluctuations. We test our analytically derived predictions on the real-world example of networks describing the soil porous architecture.

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