Geometric random intersection graphs with general connection probabilities

Abstract 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.