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.

翻译：复杂关系模式的建模已成为当代统计研究及相关数据科学领域的基石。以图形式呈现的网络为这种分析提供了天然框架。本文扩展了随机点积图（RDPG）模型以适配加权图，显著拓宽了该模型的应用范围，使其能处理边权重呈异质分布的场景。我们提出一种非参数加权（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其关联边权重分布的矩。与现有方法不同，WRDPG能够区分具有相同均值但高阶矩不同的权重分布。我们推导了基于成熟邻接谱嵌入方法改进的节点潜在位置估计量的统计保证，证明了其一致性和渐近正态性。此外，我们贡献了一个生成框架，能够生成符合（预设或数据拟合）WRDPG模型的图样本，从而便于利用恰当的参考分布对观测图指标进行分析与检验。本文系统阐述了模型定义、估计（或节点嵌入）过程及其理论保证，以及加权图生成方法，并通过可复现的示例展示了WRDPG在多种网络分析应用中的有效性。