Directed random geometric graphs: structural and spectral properties

Abstract In this work we analyze structural and spectral properties of a model directed random geometric graphs: given n vertices uniformly independently distributed on the unit square, edge is set between two if their distance smaller than connection radius ℓ , which randomly drawn from Pareto distribution. This distribution characterized by power-law decay α lower bound its support $\ell_0$?> 0 ; thus graphs depend three parameters $G(n,\alpha,\ell_0)$?> G ( n , α stretchy="false">) . By increasing for fixed $(n,\alpha)$?> transits isolated ( $\ell_0\approx 0$?> ≈ ) to complete $\ell_0 = \sqrt{2}$?> = 2 ). We first propose phenomenological expression average degree $\langle k(G) \rangle$?> fence="false" stretchy="false">⟨ k stretchy="false">⟩ works well > 3, when k self-averaging quantity. Then numerically demonstrate that V_x(G) \rangle \approx n[1-\exp(-\langle k\rangle]$?> 〈 V x 〉 stretchy="false">[ 1 − exp stretchy="false">] all where $V_x(G)$?> number nonisolated G Finally, explore use adjacency matrices represented diluted matrix ensembles; non-Hermitian Hermitian one. find good scaling parameter eigenvector mainly large