Individuals in social networks tend to exhibit homophily in their social ties, namely, they prefer bonding with others of the same social group (e.g., the same nationality, gender, socio-economic background, etc.). To formalize this phenomenon in the context of an evolving social network, we consider a mixed preferential attachment (MPA), bi-populated, network model. In this model each of the two social groups has a certain level of bias applied by its members toward other group members, ranging form homophily (affinity toward the same) to heterophily (affinity toward the different). Thus, the MPA model supports the following three simple and well-accepted observations on human behavior related to forming networks: (i) the “rich get richer” mechanism, (ii) heterogeneity and a resulting minority–majority partition, and (iii) selection bias between groups.

The main contribution of this paper is to propose a model for the evolution of a social network under these three assumptions and analyze its behavior. We show that the suggested mixed preferential attachment model exhibits, for each population, a (possibly different) power law degree distribution, i.e., a distribution where the fraction of nodes with degree $k$ is proportional to $k^{-\beta }$ (note that $\beta$ may be different for each group, as a function of the bias level and the minority–majority partition). We believe that this model may contribute to a better understanding of the forces that shape the groups and communities of our society.

社交网络中的个人倾向于表现出同质性在他们的社会关系中，即他们更喜欢与同一个社会群体的其他人（例如，相同的国籍，性别，社会经济背景等）建立联系。为了在不断发展的社交网络的背景下将此现象形式化，我们考虑了混合优先附件（MPA），双种群网络模型。在此模型中，两个社会群体中的每个群体的成员都对其他群体成员施加一定程度的偏见，范围从同质性（对相同的亲和力）到异质性（对不同的亲和力）。因此，MPA模型支持以下三个与人类形成网络有关的关于人类行为的简单且广为接受的观察结果：（i）“致富致富”机制，（ii）异质性以及由此产生的少数群体－多数群体划分，以及（iii）群体之间的选择偏见。

本文的主要贡献是在这三个假设下提出了一个社交网络演化模型，并分析了其行为。我们表明，对于每个人口，建议的混合优先依恋模型表现出一个（可能不同的）幂律度分布，即节点中具有度数的分数的分布$ķ$ 与...成正比 ${ķ}^{-\beta }$ （注意 $\beta$每个组可能会有所不同，具体取决于偏差水平和少数族裔－多数族裔划分）。我们认为，这种模式可能有助于更好地理解影响我们社会的群体和社区的力量。