A generalization of the Buckley–Osthus model for web graphs

We generalize the web graph models of Buckley and Osthus (2004) and Wu et al. (2009) to a directed version in the manner of Bollobás et al. (2003) and Cooper and Frieze (2003). The graph evolution process is a mixture of five steps. The degree distribution for this general model is derived, which generalizes the results of Buckley and Osthus (2004) and of Wu et al. (2009) to a directed version. In particular, it is proved that at the critical point, the degree sequence has the form ( 1 + O ( k − 1 / 4 log 3 / 2 k ) ) C c k ξ 2 α − 7 4 e − 2 k α .