Statistical models associated with graphs, called graphical models, have become a popular tool for representing network structures in many modern applications. Relevant features of the model are represented by vertices, edges and other higher order structures. A fundamental structural component of the network is represented by paths, which are a sequence of distinct vertices joined by a sequence of edges. The collection of all the paths joining two vertices provides a full description of the association structure between the corresponding variables. In this context, it has been shown that certain pairwise association measures can be decomposed into a sum of weights associated with each of the paths connecting the two variables. We consider a pairwise measure called an inflated correlation coefficient and investigate the properties of the corresponding path weights. We show that every inflated correlation weight can be factorized into terms, each of which is associated either to a vertex or to an edge of the path. This factorization allows one to gain insight into the role played by a path in the network by highlighting the contribution to the weight of each of the elementary units forming the path. This is of theoretical interest because, by establishing a similarity between the weights and the association measure they decompose, it provides a justification for the use of these weights. Furthermore we show how this factorization can be exploited in the computation of centrality measures and describe their use with an application to the analysis of a dietary pattern.

与图相关的统计模型称为图形模型，已成为许多现代应用程序中表示网络结构的流行工具。模型的相关特征由顶点、边等高阶结构表示。网络的基本结构组件由路径表示，路径是由一系列边连接的不同顶点序列。连接两个顶点的所有路径的集合提供了对应变量之间关联结构的完整描述。在这种情况下，已经表明某些成对关联度量可以分解为与连接两个变量的每个路径相关联的权重总和。我们考虑一种称为膨胀相关系数的成对度量，并研究相应路径权重的属性。我们表明，每个膨胀的相关权重都可以分解为术语，每个术语都与顶点或路径的边缘相关联。这种分解允许人们通过突出显示对形成路径的每个基本单元的权重的贡献来深入了解路径在网络中所扮演的角色。这是具有理论意义的，因为通过建立权重和它们分解的关联度量之间的相似性，它为使用这些权重提供了理由。此外，我们展示了如何在中心性度量的计算中利用这种分解，并描述它们在饮食模式分析中的应用。