Improving the visibility of minorities through network growth interventions

The paper investigates how to improve the network visibility of minority groups (numerically smaller groups) in growing social networks, where degree-based rankings influence access to information, social capital, and algorithmic visibility. The research question asks how two classes of interventions—group size (e.g., quotas) and behavioural (changes in homophily and mixing)—affect minority representation in top degree ranks over time, especially under preferential attachment dynamics that create persistent ranking advantages for early, well-connected nodes. The study emphasizes that homophily and group size jointly determine minority marginalisation; thus, analysing their interaction during network growth is crucial. The purpose is to provide a rigorous theoretical and computational framework to evaluate intervention effectiveness and to explore practical what-if scenarios in academia (gender parity). The importance lies in informing policies that coordinate institutional quotas with behavioural change to overcome structural, network-driven inequalities, acknowledging path dependence and ranking stability in scale-free networks.

The authors situate their work in research on social network effects on inequality and marginalisation, highlighting how homophily and group size shape visibility and access to resources. Prior studies show homophily influences minority rankings, mixing patterns affect inequality in gatherings, and preferential attachment leads to ranking stability that entrenches early advantages. Interventions have been studied for diffusion and organisational change (e.g., bridging, rewiring), yet a systematic, quantitative analysis of interventions that change network growth parameters (group sizes and mixing biases over time) has been lacking. The work builds on the Barabási–Albert (BA) preferential attachment model and the BA-Homophily extension, adding time dependence in homophily and group sizes to examine intervention effects on degree-based visibility. It also connects to empirical contexts where minorities (e.g., women in STEM) are underrepresented and to policy measures like affirmative action and quotas, noting mixed outcomes without behavioural alignment.

• Model: A two-phase extension of the BA-Homophily model of network growth. Nodes arrive sequentially and attach via preferential attachment modulated by a homophily parameter h and a minority size parameter min. Homophily h in [0,1] determines same-group (homophilic) versus cross-group (heterophilic) linking propensity; min denotes the minority fraction of arriving nodes. The process BA(h, min) is run in two phases to model an intervention: pre-intervention BA(h1, min1) for N1 arrivals followed by post-intervention BA(h2, min2) for N2 arrivals.
• Interventions: (i) Group size intervention (quota) changes min from min1 to min2. (ii) Behavioural intervention changes homophily from h1 to h2. For most analyses, phases have equal length (N1 = N2), and min2 ≥ min1, yielding final minority fraction mintotal = (min1 + min2)/2.
• Evaluation metric: Minority fraction among top 100 degree-ranked nodes (robustness in SI using top k%). Proportional representation is defined as equality between the fraction of minority in top ranks and the group’s final network-wide fraction.
• Simulation details: Pre-intervention networks with N1 ≈ 2500 nodes generated at min1 = 0.1 under heterophilic (h1 = 0.1), neutral (h1 = 0.5), or homophilic (h1 = 0.9) settings. Post-intervention continues to N ≈ 5000 with 10 replicate runs per parameter configuration. Behavioural interventions vary h2 ∈ {0.1, 0.5, 0.9} at fixed min1 = min2 = 0.1. Group-size interventions vary min2 ∈ [0.1, 0.9] with h1 = h2 fixed. Combined interventions vary h2 × min2.
• Analytical derivations: Degree growth for minority/majority in pre- and post-intervention phases derived from BA-Homophily dynamics, showing dependence on pre-intervention parameters due to preferential attachment. Post-intervention growth is analysed separately for old (pre-intervention) and new (post-intervention) nodes, explaining ranking stability and the limited impact on old high-degree nodes. Symmetry conditions identify combinations of (h2, min2) that yield similar degree growth effects for old nodes.
• Real-world case study: APS co-authorship network (1940–1970) built as a temporally growing, one-generation collaboration network; gender inferred as a binary attribute to identify women as the minority. Starting from real networks at 1940/1950/1960, hypothetical interventions are simulated for 10-year growth to the size observed in data, comparing final minority representation in top ranks to the real network. Two scenarios: fixed final minority size as in data (behavioural-only) and enforced 50% final minority size (quota) with behavioural variations.

• Behavioural interventions (vary h2 with fixed min1 = min2 = 0.1):
  • Heterophilic post-intervention behaviour (low h2) increases minority visibility in top ranks; the effect diminishes as h2 increases toward homophily.
  • Switching from homophilic (h1 = 0.9) to heterophilic (h2 low) can reverse minority under-representation to over-representation.
• Group-size interventions (vary min2 with h1 = h2):
  • Under homophilic mixing, increasing the minority quota (higher min2) raises minority representation in top ranks.
  • Under heterophilic mixing, increasing the minority quota paradoxically reduces minority representation in top ranks.
  • Under neutral mixing (h = 0.5), quotas have negligible effect on ranking representation.
• Combined interventions (vary h2 and min2):
  • A qualitative shift occurs at min2 ≈ 0.5: for min2 < 0.5, heterophilic behaviour benefits minority visibility most; for min2 > 0.5, homophilic behaviour becomes more beneficial. At min2 = 0.5, both heterophilic and homophilic outperform random attachment (h2 = 0.5).
  • The effectiveness and magnitude of improvements depend strongly on pre-intervention homophily. Homophilic initial conditions (h1 = 0.9) make proportional representation harder to achieve and generally require stronger interventions.
• Degree dynamics and ranking stability:
  • Post-intervention effects strongly impact newly arriving nodes; old high-degree nodes largely retain their positions due to preferential attachment and ranking stability (super-stable nodes), limiting the effect size on overall rankings.
  • Analytical and simulation results reveal symmetries: certain combinations of behavioural and group-size interventions can cancel or mimic each other’s effects on old nodes’ degree growth, explaining when interventions have little to no impact.
• Distance from proportional representation:
  • For homophilic pre-intervention networks, minorities remain under-represented relative to their final network size across wide parameter ranges; proportional representation is often unattainable by growth-only interventions.
  • Increasing quotas raises the final minority share but may not translate into top-rank visibility due to entrenched pre-intervention structure.
• Real-world APS case study (gender):
  • Without quotas (final minority size fixed as in data), heterophilic behaviour (h2 low) increases women’s visibility; homophilic or random behaviour reduces it.
  • With enforced 50% final minority size, the beneficial behaviour switches: homophilic behaviour (h2 high) helps women more once they are the majority among newcomers.
  • Timing matters: Early quotas (e.g., in 1940, before strong scale-free structure) improve minority visibility across behavioural regimes, as ranking stability had not yet formed.

The study shows that intervention outcomes depend on an interaction between group size dynamics and mixing behaviour. Increasing minority participation alone cannot guarantee improved visibility in degree-based rankings; coordinated behavioural change is necessary. The underlying preferential attachment mechanism locks in early advantages, making top-rank positions resistant to modification, particularly when the pre-intervention network is strongly homophilic. The observed threshold around min2 = 0.5 reflects a role reversal among newcomers: once the minority becomes the majority of incoming nodes, homophilic behaviour preferentially benefits them. Thus, policy design must consider both quotas and behavioural incentives and must tailor them to the network’s initial homophily and timing. In real collaboration networks, early implementation of quotas is especially effective, before pronounced scale-free ranking stability emerges. These insights provide guidance for designing interventions that coordinate institutional policies (e.g., hiring quotas) with behavioural strategies (e.g., promoting intergroup or intragroup collaborations) to meaningfully alter visibility in growing networks.

The paper introduces a two-phase, time-dependent BA-Homophily growth model to assess how group-size (quota) and behavioural (homophily) interventions shape minority visibility in degree rankings. It demonstrates that: (i) quotas alone may fail—and can even backfire—without appropriate behavioural alignment; (ii) the optimal behavioural strategy depends on the quota level, with a shift around min2 ≈ 0.5; (iii) initial network homophily and ranking stability constrain achievable improvements; and (iv) in real-world settings, early quotas are critical, and only specific combinations of behavioural and group-size interventions improve minority visibility. The work lays a theoretical and computational foundation for evaluating network interventions and informing policy. Future research could extend beyond binary attributes to multi-dimensional identities, incorporate additional social mechanisms (e.g., triadic closure, higher-order interactions), explore other notions of inequality beyond degree-based visibility, and develop strategies to modify existing network structure in tandem with growth interventions.

• Attribute simplification: Groups treated as binary (e.g., gender) and homophily assumed symmetric, which may not capture complex, multi-dimensional identities or asymmetric biases.
• Mechanistic scope: The model focuses on preferential attachment with homophily; it does not incorporate other micro-mechanisms such as triadic closure, community formation dynamics, or higher-order interactions.
• Visibility metric: Outcomes assessed via degree-based top-rank visibility; other dimensions of inequality (e.g., influence, access, algorithmic outcomes) are not directly evaluated.
• Parameter extremes: Simulations sometimes explore extreme quotas or homophily values that may be unrealistic but are used to reveal qualitative interactions; not intended as policy prescriptions.
• Path dependence: Strong ranking stability in scale-free networks limits the impact of growth-only interventions on established high-degree nodes, especially under homophilic initial conditions.
• Data caveats (APS case): Gender inferred from names and images reflects perceived binary gender, not self-identified gender; potential misclassification and conceptual limitations.
• Generalizability: Results are most applicable to settings where preferential attachment and homophily are dominant drivers of growth; different dynamics may yield different intervention effects.