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

CT Martinez-Martinez, JA Mendez-Bermudez, Francisco A Rodrigues, Ernesto Estrada

Submitted (2021)

Submitted (2021)

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.

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.