有向双总体偏好依恋模型及其在玻璃天花板效应分析中的应用
A Directed, Bi-Populated Preferential Attachment Model with Applications to Analyzing the Glass Ceiling Effect
摘要：
在无向网络的背景下，偏好依恋、同质性及其后果（如玻璃天花板效应）已经得到了很好的研究。然而，缺乏一个直观的，理论上易于处理的模型的有向，双填充~（即，~含有两个基团）网络，具有可变的优先连接水平，同质性和生长动力学~（例如，~新节点加入的速率、新节点主要跟随现有节点还是现有节点跟随它们等）。在很大程度上阻止了在有向网络的背景下探索这种后果，在有向网络中，由于链接的不对称性，它们更自然地发生。为此，我们提出了一个严格的理论分析的\emph{Directed Mixed Preferential Attachment}模型，并使用它来分析有向网络中的玻璃天花板效应。更具体地说，我们推导出封闭形式的表达式的幂律指数的在和出度分布的每个组~（少数和多数），并将它们相互比较，以获得见解。特别是，我们的结果产生的问题，如：\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?}的答案。这种见解揭示了结构之间的相互作用~（即，这两个群体的入度和出度分布）和动态（由同质性、偏好连接、群体规模和增长动态共同表征）。最后，我们利用所得到的分析结果来描述的条件下，玻璃天花板效应出现在一个有向网络。我们的分析结果支持详细的数值结果。
英文摘要：
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.
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