Topological versus spectral properties of random geometric graphs

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 Randic 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, $leftlangle R(G) rightrangle$ and $leftlangle H(G) rightrangle$, are highly correlated with the average number of non-isolated vertices $leftlangle V_times(G) rightrangle$; and (ii) surprisingly, the averaged--scaled Shannon entropy $leftlangle S(G) rightrangle$ is also highly correlated with $leftlangle V_times(G) rightrangle$. Therefore, we suggest that very reliable predictions of eigenvector properties of RGGs could be made by computing topological indices.