  Empirical studies show that online social networks have not only in- and
out-degree distributions with Pareto-like tails but also a high proportion of
reciprocal edges. A classical directed preferential attachment (PA) model
generates in- and out-degree distribution with power-law tails, but theoretical
properties of the reciprocity feature in this model have not yet been studied.
We derive the asymptotic results on the number of reciprocal edges between two
fixed nodes, as well as the proportion of reciprocal edges in the entire PA
network. We see that with certain choices of parameters, the proportion of
reciprocal edges in a directed PA network is close to 0, which differs from the
empirical observation. This points out one potential problem of fitting a
classical PA model to a given network dataset with high reciprocity and
indicates alternative models need to be considered.
