Directed random geometric graphs: structural and spectral properties

In this work we analyze structural and spectral properties of a model of directed random geometric graphs: given n vertices uniformly and independently distributed on the unit square, a directed edge is set between two vertices if their distance is smaller than the connection radius ℓ , which is randomly drawn from a Pareto distribution. This Pareto distribution is characterized by the power-law decay α and the lower bound of its support ℓ0 ; thus the graphs depend on three parameters G(n,α,ℓ0) . By increasing ℓ0 , for fixed (n,α) , the model transits from isolated vertices ( ℓ0≈0 ) to complete graphs ( ℓ0=2 ). We first propose a phenomenological expression for the average degree ⟨k(G)⟩ which works well for α > 3, when k is a self-averaging quantity. Then we numerically demonstrate that 〈Vx(G)〉≈n[1−exp(−〈k〉] , for all α, where Vx(G) is the number of nonisolated vertices of G. Finally, we explore the spectral properties of G(n,α,ℓ0) by the use of adjacency matrices represented by diluted random matrix ensembles; a non-Hermitian and a Hermitian one. We find that ⟨k⟩ is a good scaling parameter of spectral and eigenvector properties of G mainly for large α.