  We study spatial networks constructed by randomly placing nodes on a manifold
and joining two nodes with an edge whenever their distance is less than a
certain cutoff. We derive the general expression for the connectivity
distribution of such networks as a functional of the distribution of the nodes.
We show that for regular spatial densities, the corresponding spatial network
has a connectivity distribution decreasing faster than an exponential. In
contrast, we also show that scale-free networks with a power law decreasing
connectivity distribution are obtained when a certain information measure of
the node distribution (integral of higher powers of the distribution) diverges.
We illustrate our results on a simple example for which we present simulation
results. Finally, we speculate on the role played by the limiting case P(k)=1/k
which appears empirically to be relevant to spatial networks of biological
origin such as the ones constructed from gene expression data.
