| Full text | |
| Author(s): |
Martinez-Martinez, C. T.
;
Mendez-Bermudez, J. A.
;
Rodrigues, Francisco A.
;
Estrada, Ernesto
Total Authors: 4
|
| Document type: | Journal article |
| Source: | PHYSICAL REVIEW E; v. 105, n. 3, p. 13-pg., 2022-03-15. |
| Abstract | |
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, 0), obeying the probability density functions rho(l) and rho(theta). Here we choose rho(l) as a normal distribution with zero mean and variance sigma is an element of (0, infinity) and rho(theta) as a uniform distribution in the interval theta is an element of [0, 2 pi). 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, sigma, l), by the use of the average degree < k > and the number of nonisolated vertices V-x, 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, sigma, l)>. Then, we look for the scaling properties of the normalized average measure <(X) over bar > (where X stands for V-x, r, and S) over graph ensembles. We demonstrate that the scaling parameter of <(V-x) over bar > = <(V-x) over bar >/n is indeed < k >, with <(V-x) over bar > approximate to 1 - exp(-< k >). Meanwhile, the scaling parameter of both <(r) over bar > and <(S) over bar > is proportional to n(-gamma) < k > with gamma approximate to 0.16. (AU) | |
| FAPESP's process: | 19/23293-0 - Prediction and inference in complex systems |
| Grantee: | Francisco Aparecido Rodrigues |
| Support Opportunities: | Regular Research Grants |