Topological versus spectral properties of random geometric graphs

Abstract

In this work we perform a detailed statistical analysis of topological and spectral properties of random geometric graphs (RGGs); a graph model used to study the structure and dynamics of complex systems embedded in a two dimensional space. RGGs, $G(n,\ell )$, consist of $n$ vertices uniformly and independently distributed on the unit square, where two vertices are connected by an edge if their Euclidian distance is less or equal than the connection radius $\ell \in [0,\sqrt{2}]$. To evaluate the topological properties of RGGs we chose two well-known topological indices, the Randić index $R(G)$ and the harmonic index $H(G)$. While we characterize the spectral and eigenvector properties of the corresponding randomly-weighted adjacency matrices by the use of random matrix theory measures: the ratio between consecutive eigenvalue spacings, the inverse participation ratios and the information or Shannon entropies $S(G)$. First, we review the scaling properties of the averaged measures, topological and spectral, on RGGs. Then we show that: (i) the averaged--scaled indices, $⟨R(G)⟩$ and $⟨H(G)⟩$, are highly correlated with the average number of non-isolated vertices $⟨V_{\times }(G)⟩$; and (ii) surprisingly, the averaged--scaled Shannon entropy $⟨S(G)⟩$ is also highly correlated with $⟨V_{\times }(G)⟩$. Therefore, we suggest that very reliable predictions of eigenvector properties of RGGs could be made by computing topological indices.

8 pages, 4 figures

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