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Abstract Preferential attachment, homophily, and their consequences such as scale-free (i.e. power-law) degree distributions, the glass ceiling effect (the unseen, yet unbreakable barrier that keeps minorities and women from rising to the upper rungs of the corporate ladder, regardless of their qualifications or achievements) and perception bias are well-studied in undirected networks. However, such consequences and the factors that lead to their emergence in directed networks (e.g. author–citation graphs, Twitter) are yet to be coherently explained in an intuitive, theoretically tractable manner using a single dynamical model. To this end, we present a theoretical and numerical analysis of the novel Directed Mixed Preferential Attachment model in order to explain the emergence of scale-free degree distributions and the glass ceiling effect in directed networks with two groups (minority and majority). Specifically, we first derive closed-form expressions for the power-law exponents of the in-degree and out-degree distributions of each of the two groups and then compare the derived exponents with each other to obtain useful insights. These insights include answers to questions such as: when does the minority group have an out-degree (or in-degree) distribution with a heavier tail compared to the majority group? what factors cause the tail of the out-degree distribution of a group to be heavier than the tail of its own in-degree distribution? what effect does frequent addition of edges between existing nodes have on the in-degree and out-degree distributions of the majority and minority groups? Answers to these questions shed light on the interplay between structure (i.e. the in-degree 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 directed networks. We also provide a novel definition of the glass ceiling faced by a group via the number of individuals with large out-degree (i.e. those with many followers) normalized by the number of individuals with large in-degree (i.e. those who follow many others) and then use it to characterize the conditions that cause the glass ceiling effect to emerge in a directed network. Our analytical results are supported by detailed numerical experiments. The DMPA model and its theoretical and numerical analysis provided in this article are useful for analysing various phenomena on directed networks in fields such as network science and computational social science.
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This content will become publicly available on December 1, 2025
Mutual Information Measure for Glass Ceiling Effect in Preferential Attachment Models
This article introduces a novel mutual informationbased measure to assess the glass ceiling effect in preferential attachment networks, which advances the analysis of inequalities in attributed networks. Using Shannon entropy and generalizing to Rényi entropy, our measure evaluates the conditional probability distributions of node attributes given the node degrees of adjacent nodes, which offers a more nuanced understanding of inequality compared to traditional methods that emphasize node degree distributions and degree assortativity alone. To evaluate the efficacy of the proposed measure, we evaluate it using an analytical structural inequality model as well as historical publication data. Results show that our mutual information measure aligns well with both the theoretical model and empirical data, underscoring its reliability as a robust approach for capturing inequalities in attributed networks. Moreover, we introduce a novel stochastic optimization algorithm that utilizes a parameterized conditional logit model for edge addition. Our algorithm is shown to outperform the baseline uniform distribution based approach in mitigating the glass ceiling effect. By strategically recommending links based on this algorithm, we can effectively hinder the glass ceiling effect within networks.
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- Award ID(s):
- 2112457
- PAR ID:
- 10607958
- Publisher / Repository:
- IEEE
- Date Published:
- Journal Name:
- IEEE Transactions on Computational Social Systems
- Volume:
- 11
- Issue:
- 6
- ISSN:
- 2373-7476
- Page Range / eLocation ID:
- 7778 to 7788
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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