  Reciprocity in social networks helps understand information exchange between
two individuals, and indicates interaction patterns between pairs of users. A
recent study indicates the reciprocity coefficient of a classical directed
preferential attachment (PA) model does not match empirical evidence. In this
paper, we extend the classical 3-scenario directed PA model by adding an
additional parameter that controls the probability of creating a reciprocal
edge. Our proposed model also allows edge creation between two existing nodes,
making it a more realistic choice for fitting to real datasets. In addition to
analysis of the theoretical properties of this PA model with reciprocity, we
provide and compare two estimation procedures for the fitting of the extended
model to both simulated and real datasets. The fitted models provide a good
match with the empirical tail distributions of both in- and out-degrees. Other
mismatched diagnostics suggest that further generalization of the model is
warranted.
