A graphon is a limiting object used to describe the behaviour of large networks through a function that captures the probability of edge formation between nodes. Although the merits of graphons to describe large and unlabelled networks are clear, they traditionally are used for describing only binary edge information, which limits their utility for more complex relational data. Decorated graphons were introduced to extend the graphon framework by incorporating richer relationships, such as edge weights and types. This specificity in modelling connections provides more granular insight into network dynamics. Yet, there are no existing inference techniques for decorated graphons. We develop such an estimation method, extending existing techniques from traditional graphon estimation to accommodate these richer interactions. We derive the rate of convergence for our method and show that it is consistent with traditional non-parametric theory when the decoration space is finite. Simulations confirm that these theoretical rates are achieved in practice. Our method, tested on synthetic and empirical data, effectively captures additional edge information, resulting in improved network models. This advancement extends the scope of graphon estimation to encompass more complex networks, such as multiplex networks and attributed graphs, thereby increasing our understanding of their underlying structures.

翻译：图子（graphon）是一种极限对象，通过描述节点间连边概率的函数来刻画大型网络的行为。尽管图子在描述大型无标号网络方面的优势显而易见，但传统上仅用于描述二元连边信息，这限制了其在更复杂关系数据中的应用。装饰图子通过纳入更丰富的关系（如边权重和类型）扩展了图子框架。这种对连接建模的特异性为网络动力学提供了更精细的洞察。然而，目前尚不存在针对装饰图子的推断技术。我们发展了一种估计方法，将传统图子估计技术扩展至适应这些更丰富的交互作用。我们推导了该方法的收敛速率，并证明当装饰空间有限时，其与传统非参数理论一致。仿真实验证实这些理论速率在实践中可达。我们的方法在合成数据和实证数据上经过测试，能有效捕捉额外的边信息，从而获得改进的网络模型。这一进展将图子估计的范围扩展至涵盖更复杂的网络，如多重网络和属性图，从而增进了我们对其底层结构的理解。