Geometric random intersection graphs with general connection probabilities

Abstract

Let $\mathcal{V}$ and $\mathcal{U}$ be the point sets of two independent homogeneous Poisson processes on ${\mathbb{R}}^d$. A graph ${\mathcal{G}}_{\mathcal{V}}$ with vertex set $\mathcal{V}$ is constructed by first connecting pairs of points $(v,u)$ with $v\in \mathcal{V}$ and $u\in \mathcal{U}$ independently with probability $g(v-u)$, where $g$ is a non-increasing radial function, and then connecting two points $v_1,v_2\in \mathcal{V}$ if and only if they have a joint neighbor $u\in \mathcal{U}$. This gives rise to a random intersection graph on ${\mathbb{R}}^d$. Local properties of the graph, including the degree distribution, are investigated and quantified in terms of the intensities of the underlying Poisson processes and the function $g$. Furthermore, the percolation properties of the graph are characterized and shown to differ depending on whether $g$ has bounded or unbounded support.

References in corpus (2)

• Degree and clustering coefficient in sparse random intersection graphs
• Recurrence versus Transience for Weight-Dependent Random Connection Models