Full connectivity: corners, edges and faces

∗To whom correspondence should be addressed. We develop a cluster expansion for the probability of full connectivity of high den-sity random networks in confined geometries. In contrast to percolation phenomena at lower densities, boundary effects, which have previously been largely neglected, are not only relevant but dominant. We derive general analytical formulas that show a persis-tence of universality in a different form to percolation theory, and provide numerical confirmation. We also demonstrate the simplicity of our approach in three simple but instructive examples and discuss the practical benefits of its application to different models.