  Reciprocity is an important characteristic of directed networks and has been
widely used in the modeling of World Wide Web, email, social, and other complex
networks. In this paper, we take a statistical physics point of view and study
the limiting entropy and free energy densities from the microcanonical
ensemble, the canonical ensemble, and the grand canonical ensemble whose
sufficient statistics are given by edge and reciprocal densities. The sparse
case is also studied for the grand canonical ensemble. Extensions to more
general reciprocal models including reciprocal triangle and star densities will
likewise be discussed.
