Spectral statistics of random geometric graphs

Author： Dettmann C. P. Georgiou O. Knight G.

E-ISSN： 1286-4854|118|1|18003-18003

ISSN： 0295-5075

Source： EPL (EUROPHYSICS LETTERS), Vol.118, Iss.1, 2017-05, pp. : 18003-18003

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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 Erdős-Rényi, Barabási-Albert and Watts-Strogatz random graphs.