The master equation approach is applied to quantify structural features of growing, popularity-driven networks. The specific case of linear preferential attachment gives a non-robust degree distribution whose exponent depends on microscopic details of the network growth. This sensitivity stems from an underlying multiplicative nature of the attachment process that quantifies how the rich get richer. In spite of the non-generic features of linear preferential attachment, this mechanism seems to account for the entire Physical Review citation data. Some peculiar features of superlinear attachments rates will also be discussed. Finally, a model with random growth that is augmented by redirection and and by copying will be discussed.
The former gives a power-law degree distribution with a non-universal exponent. The latter leads radically different in- and out-degree distributions.