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,), 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 than or equal to the connection radius ∈[0,√2]. To evaluate the topological properties of RGGs we chose two well-known topological indices, the Randi {c} \'{c}{} index $R(G)$ and the harmonic index $H(G)$. 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 nonisolated vertices V_×(G); and (ii) surprisingly, the averaged-scaled Shannon entropy S(G) is also highly correlated with V_×(G). Therefore, we suggest that very reliable predictions of eigenvector properties of RGGs could be made by computing topological indices.