Fig. 3: The minority (dis)advantage in networks.

a The distance of groups' average degree to the overall average degree (z score) with different minority fraction f₀ and at varying levels of symmetrical mixing (i.e., h₀₀ = h₁₁ = h_rr). The minority members have a degree advantage or disadvantage if the system is, respectively, at a heterophilic (h_rr < 0.5) or homophilic (h_rr > 0.5) regime. b The distance of the minority group degree to the overall average degree, denoted k₀ − 〈k〉, at different levels of asymmetrical mixing (i.e., h₀₀ ≠ h₁₁). The majority mixing h₁₁ explains much of the variance of k₀ − 〈k〉. c The variation of k₀ − 〈k〉 with changes in the minority mixing h₀₀, with fixed h₁₁ = 0.5 and different minority fraction f₀. The minority mixing can have opposite impacts on degree inequality depending on the minority fraction, which suggests a qualitative transition in the system. d The derivative of k₀ − 〈k〉 as a function of f₀. The zero of this function represents the critical minority fraction, denoted \({f}_{0}^{* }\), at which the qualitative transition occurs. In the plot, same-color curves represent varying levels of minority mixing, from h₀₀ = 0 to h₀₀ = 1. e Two regimes delineated by \({f}_{0}^{* }\), with initial h₀₀ = 0.5. These regimes mean that the minority group degree may either increase or decrease with a raise of h₀₀. f The parameter space of critical minority fraction. Given h₀₀, the upper limit of \({f}_{0}^{* }\), denoted \(\overline{{f}_{0}^{* }}\) (dashed line), represents the smallest minority size allowing higher minority homophily without decreasing group average degree, regardless of the majority mixing.