Connectivity Distribution of Spatial Networks

Details Connectivity Distribution of Spatial Networks Connectivity Distribution of Spatial Networks

2003-02-26

arxiv.org/abs/cond-mat/0302544v1

Phys. Rev. E 68 (2003) 026128

Physics

Condensed Matter

Disordered Systems and Neural Networks

6 pages, 1 figure

Scientific paper

10.1103/PhysRevE.68.026128

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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