标题： 加权随机点积图 显示英文标题

标题： Weighted Random Dot Product Graphs

摘要： 复杂关系模式的建模已成为当代统计研究和相关数据科学领域的基石。网络，表示为图，为这种分析提供了自然的框架。本文扩展了随机点积图（RDPG）模型以适应加权图，显著扩大了该模型的应用范围，适用于边具有异质权重分布的场景。我们提出了一种非参数加权（W）RDPG模型，为每个节点分配一系列潜在位置。这些节点向量的内积通过矩生成函数指定了它们相邻边权重分布的矩。通过这种方式，并且与先前的工作不同，WRDPG能够区分具有相同均值但其他高阶矩不同的权重分布。我们推导了从核心邻接谱嵌入工作中改编的节点潜在位置估计器的统计保证，确立了其一致性和渐近正态性。我们还贡献了一个生成框架，使能够采样符合（指定或拟合数据的）WRDPG的图，从而促进了例如使用审慎参考分布来分析和检验观测图指标。本文组织结构旨在形式化模型定义、估计（或节点嵌入）过程及其保证，以及生成加权图的方法，并辅以说明性和可再现的例子展示了WRDPG在各种网络分析应用中的有效性。 显示英文摘要

摘要： Modeling of intricate relational patterns has become a cornerstone of contemporary statistical research and related data science fields. Networks, represented as graphs, offer a natural framework for this analysis. This paper extends the Random Dot Product Graph (RDPG) model to accommodate weighted graphs, markedly broadening the model's scope to scenarios where edges exhibit heterogeneous weight distributions. We propose a nonparametric weighted (W)RDPG model that assigns a sequence of latent positions to each node. Inner products of these nodal vectors specify the moments of their incident edge weights' distribution via moment-generating functions. In this way, and unlike prior art, the WRDPG can discriminate between weight distributions that share the same mean but differ in other higher-order moments. We derive statistical guarantees for an estimator of the nodal's latent positions adapted from the workhorse adjacency spectral embedding, establishing its consistency and asymptotic normality. We also contribute a generative framework that enables sampling of graphs that adhere to a (prescribed or data-fitted) WRDPG, facilitating, e.g., the analysis and testing of observed graph metrics using judicious reference distributions. The paper is organized to formalize the model's definition, the estimation (or nodal embedding) process and its guarantees, as well as the methodologies for generating weighted graphs, all complemented by illustrative and reproducible examples showcasing the WRDPG's effectiveness in various network analytic applications.