Topological versus spectral properties of random geometric graphs
Rocio Aguilar-Sánchez, Jose Antonio Mendez-Bermudez, Francisco A. Rodrigues, José M. Sigarreta · Physical review. E · 2020
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, $G(n,\ensuremath{\ell})$, 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 than or equal to the connection radius $\ensuremath{\ell}\ensuremath{\in}[0,\sqrt{2}]$. To evaluate the topological properties of RGGs we chose two well-known topological indices, the Randi\ifmmode \acute{c}\else \'{c}\fi{} 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, $\ensuremath{\langle}R(G)\ensuremath{\rangle}$ and $\ensuremath{\langle}H(G)\ensuremath{\rangle}$, are highly correlated with the average number of nonisolated vertices $\ensuremath{\langle}{V}_{\ifmmode\times\else\texttimes\fi{}}(G)\ensuremath{\rangle}$; and (ii) surprisingly, the averaged-scaled Shannon entropy $\ensuremath{\langle}S(G)\ensuremath{\rangle}$ is also highly correlated with $\ensuremath{\langle}{V}_{\ifmmode\times\else\texttimes\fi{}}(G)\ensuremath{\rangle}$. Therefore, we suggest that very reliable predictions of eigenvector properties of RGGs could be made by computing topological indices.