  Many complex networks exhibit a percolation transition involving a
macroscopic connected component, with universal features largely independent of
the microscopic model and the macroscopic domain geometry. In contrast, we show
that the transition to full connectivity is strongly influenced by details of
the boundary, but observe an alternative form of universality. Our approach
correctly distinguishes connectivity properties of networks in domains with
equal bulk contributions. It also facilitates system design to promote or avoid
full connectivity for diverse geometries in arbitrary dimension.
