Spectral statistics of random geometric graphs

Abstract

We use random matrix theory to study the spectrum of random geometric graphs, a fundamental model of spatial networks. Considering ensembles of random geometric graphs we look at short range correlations in the level spacings of the spectrum via the nearest neighbour and next nearest neighbour spacing distribution and long range correlations via the spectral rigidity Delta_3 statistic. These correlations in the level spacings give information about localisation of eigenvectors, level of community structure and the level of randomness within the networks. We find a parameter dependent transition between Poisson and Gaussian orthogonal ensemble statistics. That is the spectral statistics of spatial random geometric graphs fits the universality of random matrix theory found in other models such as Erdos-Renyi, Barabasi-Albert and Watts-Strogatz random graph.

arXiv comment19 pages, 6 figures. Substantially updated from previous version

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• Geometrical and spectral study of $\beta$-skeleton graphs
• Eigenvalue ratio statistics of complex networks: Disorder vs. Randomness
• Isotropic random geometric networks in two dimensions with a penetrable cavity
• Random geometric complexes and graphs on Riemannian manifolds in the thermodynamic limit
• Large algebraic connectivity fluctuations in spatial network ensembles imply a predictive advantage from node location information
• A class of 2x2 correlated random-matrix models with Brody spacing distribution
• Random Matrix Analysis of Multiplex Networks