Inequalities in social networks arise from linking mechanisms, such as preferential attachment (connecting to popular nodes), homophily (connecting to similar others), and triadic closure (connecting through mutual contacts). Preferential attachment drives degree inequality and homophily drives segregation, but we know less about how these two mechanisms interact with triadic closure.

This gap limits our understanding of how network inequalities emerge. We introduce PATCH, a network growth model that combines all three mechanisms, and use it to study how they create disparities within and between two groups in undirected networks. Simulations show that homophily and preferential attachment increase segregation and degree inequality. Triadic closure has varied effects: conditional on the other mechanisms, it increases population-wide degree inequality while reducing segregation and between-group degree disparities.

We demonstrate PATCH’s explanatory potential using fifty years of Physics and Computer Science collaboration and citation networks exhibiting persistent gender disparities. PATCH reproduces these gender disparities when it combines preferential attachment, moderate gender homophily, and triadic closure.

By connecting mechanisms to observed inequalities, PATCH shows how their interplay sustains group disparities and how improving one inequality dimension may affect others.

J. Bachmann, S. Martin-Gutierrez, L. Espín-Noboa, N. Cinardi, F. Karimi, Network inequality through preferential attachment, triadic closure, and homophily, Scientific Reports (2026).