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。以图形式表示的网络为此类分析提供了自然框架。本文扩展了随机点积图模型，使其能够处理加权图，显著拓宽了该模型在边权呈现异质分布场景下的适用范围。我们提出了一种非参数化加权随机点积图模型，该模型为每个节点分配一组潜在位置序列。通过矩生成函数，这些节点向量的内积可确定其关联边权分布的各阶矩。与现有技术不同，WRDPG 因此能够区分具有相同均值但高阶矩不同的权重分布。基于经典邻接谱嵌入方法，我们推导了节点潜在位置估计量的统计保证，证明了其一致性与渐近正态性。同时，我们提出了一种生成框架，能够对符合（预设或数据拟合的）WRDPG 的图进行采样，从而为使用合理参考分布分析和检验观测图度量提供了便利。本文通过形式化定义模型、阐述估计（或节点嵌入）过程及其理论保证、提出加权图生成方法，并结合展示 WRDPG 在各种网络分析应用中有效性的可复现示例，系统性地呈现了研究成果。