Geometric random intersection graphs with general connection probabilities

Let V and U be the point sets of two independent homogeneous Poisson processes on Rᵈ . A graph GV with vertex set V is constructed by first connecting pairs of points ( v , u ) with v and u independently with probability $g(v-u)$ , where g is a non-increasing radial function, and then connecting two points v₁,v₂ if and only if they have a joint neighbor u . This gives rise to a random intersection graph on Rᵈ . 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.