Multispecies grand-canonical models for networks with reciprocity

Reciprocity is a second-order correlation that has been recently detected in all real directed networks and shown to have a crucial effect on the dynamical processes taking place on them. However, no current theoretical model generates networks with this nontrivial property. Here we propose a grand-canonical class of models reproducing the observed patterns of reciprocity by regarding single and double links as Fermi particles of different chemical species governed by the corresponding chemical potentials. Within this framework we find interesting special cases such as the extensions of random graphs, the configuration model, and hidden-variable models. Our theoretical predictions are also in excellent agreement with the empirical results for networks with well-studied reciprocity.