Non-uniform random graphs on the plane: A scaling study

Abstract

We consider random geometric graphs on the plane characterized by a non-uniform density of vertices. In particular, we introduce a graph model where n vertices are independently distributed in the unit disc 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 an uniform distribution in the interval θ ∈ [0, 2π). Then, two vertices are connected by an edge if their Euclidian distance is less or equal than 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 non-isolated 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.