Improving the visibility of minorities through network growth interventions

Historical disadvantages often marginalize minority groups, and network structure plays a crucial role in perpetuating these inequalities. Minority group members' network positions significantly influence their access to information, social capital, and visibility in algorithmic rankings. However, social networks evolve over time due to institutional interventions, societal changes, and behavioral shifts, potentially increasing the representation of previously underrepresented groups. Examples include affirmative action policies in academia and laws favoring women in corporate boards. These changes, termed "group size interventions," are insufficient to address structural inequalities without considering network structure and group mixing biases. Homophily, the tendency for similar individuals to connect, often reinforces existing inequalities. Even with increased group size, homophily can hinder minority representation in rankings, as demonstrated by studies on women in citation networks. Therefore, it is essential to also investigate "behavioral interventions," which involve changes in individuals' connection preferences. Behavioral interventions can manifest through various mechanisms, such as changes in organizational hiring practices or the formation of support networks within minority communities. However, behavioral changes alone may not suffice without institutional enforcement. Existing research lacks a rigorous quantitative investigation of the interplay between group size and behavioral interventions in overcoming structural barriers and marginalization. This research focuses on network-based inequalities, using degree centrality as a measure of social capital and visibility. Minority representation in top centrality ranks is a function of group size and homophily. Homophily restricts minority connections to the majority, leading to underrepresentation in degree-based rankings. Most models of growing social networks assume constant group mixing biases, neglecting the time-dependent nature of interactions. In large-scale networks, preferential attachment creates a "rich-get-richer" effect, making it difficult for minorities to attain high ranks. This study addresses this gap by considering both group size changes and behavioral changes within a network growth model. The goal is to understand how network positions are formed and identify effective interventions to improve the position of disadvantaged groups.

The paper draws upon existing literature demonstrating the link between network structure and social inequality (DiMaggio & Garip, 2012; Karimi et al., 2022), focusing on how network position impacts access to resources and visibility in algorithmic rankings (Karimi et al., 2018; Espin-Noboa et al., 2022). It acknowledges the historical underrepresentation of minorities in STEM fields (National Science Foundation, 2015; Eccles, 1989; Armstrong & Jovanovic, 2015) and the use of interventions like scholarship programs and affirmative action. The role of homophily in shaping network structure and inequality is discussed (McPherson et al., 2001), along with its impact on minority group visibility (Lerman et al., 2016; Lee et al., 2019). The paper also cites research showing how group size changes can influence network dynamics (Goodreau et al., 2009; Álvarez Rivadulla et al., 2022; Dennissen et al., 2019; Oliveira et al., 2022; Nettasinghe et al., 2021) and the importance of considering time dependencies in interactions (Mucha et al., 2010; Scholtes et al., 2014; Neuhäuser et al., 2021). The limitations of interventions focusing solely on network structure (Valente, 2012; Valente & Fujimoto, 2010; Centola, 2021) are also highlighted, emphasizing the need for computational models and quantitative methods to evaluate the effectiveness of different intervention strategies (Broader scope is key to the future of ‘science of science’, 2022). The study builds upon the Barabási-Albert (BA) preferential attachment model (Barabási & Albert, 1999) and its extension, the BA-Homophily model (Karimi et al., 2018), to create a more nuanced model incorporating time-dependent parameters.

The authors develop a two-phase network growth model extending the BA-Homophily model. This model incorporates time-dependent homophily and minority group size parameters. The first phase represents the pre-intervention network formation, characterized by initial homophily (h₁) and minority group size (min₁). The second phase simulates the impact of interventions, with altered homophily (h₂) and minority group size (min₂). The model allows for two types of interventions: 1. **Group size interventions:** These interventions change the proportion of minority nodes among newly arriving nodes, simulating quotas or similar policies. This is achieved by changing the parameter min during the post-intervention growth phase. 2. **Behavioral interventions:** These interventions change the homophily parameter (h), reflecting a shift in individuals' connection preferences (how likely individuals are to connect with nodes from the same or different groups). This change is implemented by changing the parameter h during the second growth phase. The model generates networks with approximately 5000 nodes, with half added in each phase. The authors analyze the minority fraction within the top 100 ranked nodes (based on degree centrality) to assess the effect of the interventions. They examine the effects of group size and behavioral interventions both individually and in combination, varying the parameters systematically and averaging the results over multiple simulations (10 simulations). Robustness checks using the top k% of nodes (supplementary figure 4) are also performed. They analyze the degree distributions and growth in the pre- and post-intervention networks to understand the mechanisms behind the observed changes. The analytical derivations help in identifying combinations of interventions with the most impactful effects. Finally, they apply their model to a real-world case study examining gender representation in the American Physical Society (APS) co-authorship network from 1940-1970, simulating interventions to assess the effectiveness of potential strategies in achieving gender parity. The gender attribute was inferred from author names using an algorithm combining name-based inference with facial detection based on Google image search results, using methodology similar to Kong et al. (2022), acknowledging the limitations of this method in capturing non-binary genders and fluid gender identities. The paper provides detailed mathematical formulations for the network model, the calculation of growth exponents in both phases, and analyses of the degree dynamics under varying intervention parameters, including the influence of pre-existing network structure. The authors use analytical derivations to explain the observed interaction effects between the group size and behavioural interventions.

The study's key findings highlight the complex interplay between group size and behavioral interventions in influencing minority group visibility in networks: 1. **Interdependence of Interventions:** The study demonstrates that group size and behavioral interventions are interdependent. Simply increasing the size of a minority group (e.g., through quotas) does not guarantee increased visibility in network rankings. Effective interventions require a coordinated approach, considering both group size and behavioral changes. The type of intervention that best improves a minority's visibility is dependent on the minority group's size. 2. **Impact of Homophily:** The impact of interventions strongly depends on the level of homophily (the tendency for like-to-connect) in the pre-intervention network. In highly homophilic networks (where individuals prefer to connect with others like themselves), even extreme group size interventions are less effective in improving minority visibility. The structural inequalities established early in the network's growth are difficult to overcome solely by adjusting future network growth. This is due to the high degree of stability in the rankings of high-degree nodes, established in the preferential attachment based growth process of the Barabási-Albert model. 3. **Behavioral Intervention Strategies:** The optimal behavioral intervention strategy depends on the final minority group size after the intervention. For smaller final minority sizes, heterophilic behavior (where minority and majority nodes are more likely to connect) is more beneficial. For larger final minority sizes (where the minority represents more than half of the nodes arriving after the intervention), homophilic behavior (where minority nodes predominantly connect with other minority nodes) tends to improve visibility. 4. **Real-world Application (APS Network):** The case study using the APS co-authorship network showed that achieving greater gender parity necessitates considering both group size and behavioral interventions. The effectiveness of the interventions is highly dependent on the timing of implementation. Early implementations of quotas can be highly beneficial even without strong behavioral changes. This effect is less pronounced for pure behavioral interventions. 5. **Analytical Support:** The study provides analytical derivations that explain the observed effects, particularly regarding the degree growth of nodes in both the pre- and post-intervention phases. The analytical derivations support the empirically observed impact of the interventions. They show that certain combinations of interventions have little to no effect on the visibility of the minority in the network. They also support the observation that the pre-existing network structure highly influences the success of intervention policies.

This study's findings significantly advance our understanding of how to effectively address network-based inequalities. The emphasis on the interdependence of group size and behavioral interventions highlights the limitations of single-faceted approaches. The dependence on pre-existing network conditions underscores the importance of early intervention and addressing the initial conditions to truly equalize visibility in the network. The results suggest that policies aimed at increasing minority representation should not only focus on increasing minority group size but also on fostering inclusive environments that encourage heterophilic connections initially, and homophilic connection once the minority group's size becomes sufficiently large to enable the minority group to build a cumulative advantage. The study's analytical framework offers a valuable tool for predicting the effectiveness of various intervention strategies under different network conditions. The real-world application to the APS network further validates the model's applicability to real-world social systems and demonstrates how these findings can inform policy decisions in diverse settings. The study's limitations, discussed further below, provide guidance for future research.

This research provides a novel theoretical and computational framework for analyzing the effectiveness of interventions to improve minority group visibility in networks. The key finding is the interdependence of group size and behavioral interventions, highlighting the need for coordinated strategies. The model's application to a real-world dataset demonstrates its practical relevance. Future research could explore the dynamics of interventions in more complex network structures, investigate additional centrality measures beyond degree centrality, and examine the impact of interventions on other aspects of inequality beyond network visibility.

The study's reliance on a simplified model may not fully capture the complexity of real-world social dynamics, such as the influence of higher-order network structures or the presence of multiple group memberships. The binary nature of the group attribute may not fully capture the nuances of real-world social identities. The method used for gender inference in the APS network data has limitations in accurately capturing non-binary genders and fluid gender identities. While the analytical derivations provide valuable insights, they rely on some approximations, potentially affecting the precision of the results. Future research could explore more detailed models incorporating additional factors to improve the model's accuracy and generalizability.