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Connectivity Distribution of Spatial Networks

Carl Herrmann·Marc Barthelemy·Paolo Provero
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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)=1/k which appears empirically to be relevant to spatial networks of biological origin such as the ones constructed from gene expression data.

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Carl Herrmann, Marc Barthelemy, Paolo Provero. 2003-02-26. Connectivity Distribution of Spatial Networks. https://doi.org/10.1103/physreve.68.026128

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