Preferential attachment is an appealing mechanism for modeling power-law
behavior of the degree distributions in directed social networks. In this
paper, we consider methods for fitting a 5-parameter linear preferential model
to network data under two data scenarios. In the case where full history of the
network formation is given, we derive the maximum likelihood estimator of the
parameters and show that it is strongly consistent and asymptotically normal.
In the case where only a single-time snapshot of the network is available, we
propose an estimation method which combines method of moments with an
approximation to the likelihood. The resulting estimator is also strongly
consistent and performs quite well compared to the MLE estimator. We illustrate
both estimation procedures through simulated data, and explore the usage of
this model in a real data example. At the end of the paper, we also present a
semi-parametric method to model heavy-tailed features of the degree
distributions of the network using ideas from extreme value theory.