Nonuniform random graphs on the plane: A scaling study

We consider random geometric graphs on the plane characterized by a nonuniform density of vertices. In particular, we introduce a graph model where n vertices are independently distributed in the unit disk with positions, in polar coordinates (l,θ), obeying the probability density functions ρ(l) and ρ(θ). Here we choose ρ(l) as a normal distribution with zero mean and variance σ∈(0,∞) and ρ(θ) as a uniform distribution in the interval θ∈[0,2π). Then, two vertices are connected by an edge if their Euclidean distance is less than or equal to the connection radius ℓ. We characterize the topological properties of this random graph model, which depends on the parameter set (n,σ,ℓ), by the use of the average degree ⟨k⟩ and the number of nonisolated vertices V×, while we approach their spectral properties with two measures on the graph adjacency matrix: the ratio of consecutive eigenvalue spacings r and the Shannon entropy S of eigenvectors. First we propose a heuristic expression for ⟨k(n,σ,ℓ)⟩. Then, we look for the scaling properties of the normalized average measure ⟨¯¯¯X⟩ (where X stands for V×, r, and S) over graph ensembles. We demonstrate that the scaling parameter of ⟨¯¯¯¯¯V×⟩=⟨V×⟩/n is indeed ⟨k⟩, with ⟨¯¯¯¯¯V×⟩≈1−exp(−⟨k⟩). Meanwhile, the scaling parameter of both ⟨¯r⟩ and ⟨¯¯¯S⟩ is proportional to n−γ⟨k⟩ with γ≈0.16.J.A.M.-B. acknowledges financial support from CONACyT (Grant No. A1-S-22706) and BUAP (Grant No. 100405811-VIEP2021). E.E. thanks financial support from Ministerio de Ciencia, Innovacion y Universidades, Spain, for Grant No. PID2019-107603GB-I00, “Hubs-repelling/attracting Laplacian operators and related dynamics on graphs/networks.” F.A.R. acknowledges CNPq (Grant No. 309266/2019-0) and FAPESP (Grant No. 19/23293-0) for the financial support given for this research.Peer reviewe