  Preferential attachment, homophily and, their consequences such as the glass
ceiling effect have been well-studied in the context of undirected networks.
However, the lack of an intuitive, theoretically tractable model of a directed,
bi-populated~(i.e.,~containing two groups) network with variable levels of
preferential attachment, homophily and growth dynamics~(e.g.,~the rate at which
new nodes join, whether the new nodes mostly follow existing nodes or the
existing nodes follow them, etc.) has largely prevented such consequences from
being explored in the context of directed networks, where they more naturally
occur due to the asymmetry of links. To this end, we present a rigorous
theoretical analysis of the \emph{Directed Mixed Preferential Attachment} model
and, use it to analyze the glass ceiling effect in directed networks. More
specifically, we derive the closed-form expressions for the power-law exponents
of the in- and out- degree distributions of each group~(minority and majority)
and, compare them with each other to obtain insights. In particular, our
results yield answers to questions such as: \emph{when does the minority group
have a heavier out-degree (or in-degree) distribution compared to the majority
group? what effect does frequent addition of edges between existing nodes have
on the in- and out- degree distributions of the majority and minority groups?}.
Such insights shed light on the interplay between the structure~(i.e., the in-
and out- degree distributions of the two groups) and dynamics~(characterized
collectively by the homophily, preferential attachment, group sizes and growth
dynamics) of various real-world networks. Finally, we utilize the obtained
analytical results to characterize the conditions under which the glass ceiling
effect emerge in a directed network. Our analytical results are supported by
detailed numerical results.
