title: Connectivity distribution of spatial networks.
abstract: 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) proportional, variant k(-1) which appears empirically to be relevant to spatial networks of biological origin such as the ones constructed from gene expression data.
authors.affiliation: Dipartimento di Fisica Teorica dell'Università di Torino and INFN, Sezione di Torino, Via P. Giuria 1, 10125 Torino, Italy.
authors.forename: Carl
authors.initials: C
authors.lastname: Herrmann
authors.forename: Marc
authors.initials: M
authors.lastname: Barthélemy
authors.forename: Paolo
authors.initials: P
authors.lastname: Provero