  We address the connectivity of large-scale ad hoc heterogeneous wireless
networks, where secondary users exploit channels temporarily unused by primary
users and the existence of a communication link between two secondary users
depends on not only the distance between them but also the transmitting and
receiving activities of nearby primary users. We introduce the concept of
connectivity region defined as the set of density pairs -- the density of
secondary users and the density of primary transmitters -- under which the
secondary network is connected. Using theories and techniques from continuum
percolation, we analytically characterize the connectivity region of the
secondary network and reveal the tradeoff between proximity (the number of
neighbors) and the occurrence of spectrum opportunities. Specifically, we
establish three basic properties of the connectivity region -- contiguity,
monotonicity of the boundary, and uniqueness of the infinite connected
component, where the uniqueness implies the occurrence of a phase transition
phenomenon in terms of the almost sure existence of either zero or one infinite
connected component; we identify and analyze two critical densities which
jointly specify the profile as well as an outer bound on the connectivity
region; we study the impacts of secondary users' transmission power on the
connectivity region and the conditional average degree of a secondary user, and
demonstrate that matching the interference ranges of the primary and the
secondary networks maximizes the tolerance of the secondary network to the
primary traffic load. Furthermore, we establish a necessary condition and a
sufficient condition for connectivity, which lead to an outer bound and an
inner bound on the connectivity region.
