We develop an approach to generate random graphs to a target level of assortativity by using copula structures in graphons. Unlike existing random graph generators, we do not use rewiring or binning approaches to generate the desired random graph. Instead, we connect Archimedean bivariate copulas to graphons in order to produce flexible models that can generate random graphs to target assortativity. We propose three models that use the copula distribution function, copula density function and their mixed tensor product to produce networks. We express the assortativity coefficient in terms of homomorphism densities. Establishing this relationship forges a connection between the parameter of the copula and the frequency of subgraphs in the generated network. Therefore, our method attains a desired the subgraph distribution as well as the target assortativity. We establish the homomorphism densities and assortativity coefficient for each of the models. Numerical examples demonstrate the ability of the proposed models to produce graphs with different levels of assortativity.

翻译：我们提出了一种利用图函数中的Copula结构生成具有目标同配性水平的随机图的方法。与现有的随机图生成方法不同，我们既不采用重连策略，也不使用分箱方法来生成目标随机图。相反，我们通过将阿基米德二元Copula与图函数相结合，构建了能够灵活生成具有目标同配性随机图的模型。我们提出了三种模型，分别利用Copula分布函数、Copula密度函数及其混合张量积来生成网络。我们将同配系数表示为同态密度的函数。这一关系的建立将Copula参数与生成网络中子图的出现频率联系起来。因此，我们的方法既能实现期望的子图分布，又能达到目标同配性水平。我们为每个模型推导了同态密度与同配系数的表达式。数值实验证明了所提模型能够生成具有不同同配性水平的图。