Figure 2

The analytical and numerical exponent of the degree distribution for the minority (A) and majority (B) as a function of homophily, for various minority sizes. The degree distributions follow a power-law p(k) ∝ k^γ in which the exponent of the distribution (γ) depends on homophily (h) and the minority fraction (shown by different colors). The dashed lines are the expected degree exponents given by our analytical derivation (see Methods) and the dots represent the fitted value from the simulations of over 5,000 nodes. The analytical results are in excellent agreement with simulation. The minority fraction ranges from 0.05 to 0.5. For minority nodes (A), in the heterophilic regime (h < 0.5), the degree exponent ranges from −2 to −3, which shows the advantage these nodes have as their degrees grow to large values. In the homophilic regime (h > 0.5), the exponent shows a non-linear behaviour: first the degree exponent decreases, which means that degree growth for the minority nodes becomes limited, and they are thus less well-connected. However, this effect is compensated in high homophilic regime by in-group support, which explains why the exponent increases for h > 0.8. For majority nodes (B), in the heterophilic situation the growth of their degree is limited, in particular for small minority fractions. In the homophilic regime, the exponent of the majority degree always remains close to −3: the majority nodes do not gain extra advantage due to the size of their group.