Scale-free networks embedded in fractal space

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.