  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.

Statistical Mechanics;Social and Information Networks;Physics and Society