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

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 hki 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.