Preferential attachment is an appealing mechanism for modeling power-law behavior of degree distributions in social networks.  In this talk, we consider fitting a directed linear preferential attachment model to network data under three data scenarios: 1) When the full history of the network growth is given, MLE of the parameter vector and its asymptotic properties are derived.  2) When only a single-time snapshot of the network is available, an estimation method combining method of moments with an approximation to the likelihood is proposed.  3) When the data are believed to have come from a misspecified model or have been corrupted, a semi-parametric approach to model heavy-tailed features of the degree distributions is presented, using ideas from extreme value theory.  We illustrate these estimation procedures and explore the usage of this model through simulated and real data examples.