Isotropic random geometric networks in two dimensions with a penetrable cavity

Saha, Dipa; Mitra, Sayantan; Bhowmik, Bishnu; Sensharma, Ankur

doi:10.1016/j.physa.2021.126297

Cited by 2 publications

(1 citation statement)

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“…Also, recently, RGGs over more intricate bounding domains (i.e. elongated rectangles [9][10][11] or annuli [12,13]) have been studied. RGGs with non-uniform vertex distributions have been explored too [14][15][16]; while quasirandom vertex distributions have also been considered in [17].…”

Section: Introductionmentioning

confidence: 99%

Directed random geometric graphs: structural and spectral properties

Peralta-Martinez

Méndez-Bermúdez

2022

J. Phys. Complex.

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In this work we analyze structural and spectral properties of a model of directed random geometric graphs: Given $n$ vertices uniformly and independently distributed on the unit square, a directed edge is set between two vertices if their distance is smaller than the connection radius $\ell$, which is randomly drawn from a Pareto distribution. This Pareto distribution is characterized by the power-law decay $\alpha$ and the lower bound of its support $\ell_0$; thus the graphs depend on three parameters $G(n,\alpha,\ell_0)$. By increasing $\ell_0$, for fixed $(n,\alpha)$, the model transits from isolated vertices ($\ell_0\approx 0$) to complete graphs ($\ell=\sqrt{2}$). We first propose a phenomenological expression for the average degree $\langle k(G) \rangle$ which works well for $\alpha>3$, when $k$ is a self-averaging quantity. Then we demonstrate that $\langle V_x(G) \rangle \approx n[1-\exp(-\langle k\rangle]$, for all $\alpha$, where $V_x(G)$ is the number of nonisolated vertices of $G$. Finally, we explore the spectral properties of $G(n,\alpha,\ell_0)$ by the use of adjacency matrices represented by diluted random matrix ensembles; a non-Hermitian and a Hermitian one. We find that $\langle k \rangle$ is a good scaling parameter of spectral and eigenvector properties of $G$ mainly for large $\alpha$.

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