标题： 装饰图的推断及其在多路网络中的应用 显示英文标题

标题： Inference for decorated graphs and application to multiplex networks

摘要： 图是一种极限对象，用于通过一个函数来描述大型网络的行为，该函数捕捉节点之间形成边的概率。 尽管图在描述大型和无标签网络方面的优点显而易见，但它们传统上仅用于描述二元边信息，这限制了它们在更复杂关系数据中的实用性。 装饰图被引入以通过结合更丰富的关系（如边权重和类型）来扩展图框架。 这种对连接的建模特异性提供了对网络动态的更细致见解。 然而，目前尚无针对装饰图的现有推断技术。 我们开发了这样的估计方法，将传统的图估计现有技术扩展以适应这些更丰富的交互作用。 我们推导了我们方法的收敛速率，并证明当装饰空间是有限时，它与传统非参数理论一致。 模拟结果证实了这些理论速率在实践中得以实现。 我们的方法在合成和实证数据上进行了测试，有效捕获了额外的边信息，从而改进了网络模型。 这一进展将图估计的范围扩展到涵盖更复杂的网络，如多路网络和属性图，从而增强了我们对它们底层结构的理解。 显示英文摘要

摘要： 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.