A Directed, Bi-Populated Preferential Attachment Model with Applications to Analyzing the Glass Ceiling Effect

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.