Connectivity Distribution of Spatial Networks

TL;DR

This paper derives the connectivity distribution for spatial networks formed by randomly placed nodes, revealing conditions for exponential and scale-free distributions, with implications for biological networks.

Contribution

It provides a general analytical framework for the connectivity distribution in spatial networks based on node placement and explores conditions leading to different distribution types.

Findings

Regular densities lead to faster-than-exponential decay in connectivity.

Divergence of a specific information measure yields scale-free networks.

Empirical relevance of P(k)=1/k in biological 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)=1/k which appears empirically to be relevant to spatial networks of biological origin such as the ones constructed from gene expression data.