Although there is much research on network formation based on the preferential attachment rule, the research did not come up with a formula that, on the one hand, can reproduce shapes of cumulative degree distributions of empirical complex networks and, on the other hand, can represent intuitively theories on individual behavior. In this paper, we propose a formula that closes this gap by integrating into the formula for the preferential attachment rule (i.e., a node with higher degree is more likely to gain a new link) a representation of the theory of individual behavior with respect to nodes preferring to connect to other nodes with similar attributes (i.e., homophily). Based on this formula, we simulate the shapes of cumulative degree distributions for different levels of homophily and five different seed networks. Our simulation results suggest that homophily and the preferential attachment rule interact for all five types of seed networks. Surprisingly, the resulting cumulative degree distribution in log-log scale always shifts from a concave shape to a convex shape, as the level of homophily gets larger. Therefore, our formula can explain intuitively why some of the empirical complex networks show a linear cumulative degree distribution in log-log scale while others show either a concave or convex shape. Furthermore, another major finding indicates that homophily makes people of a group richer than people outside this group, which is a surprising and significant finding.
Kim, K., & Altmann, J. (2017). Effect of homophily on network formation. Communications in Nonlinear Science and Numerical Simulation, 44, 1339–1351. https://doi.org/10.1016/j.cnsns.2016.08.011