# Path Weights in Concentration Graphs

## TL;DR

This paper explores the properties and interpretations of path weights in concentration graphs, extending covariance decomposition to other association measures and illustrating their application in dietary network analysis.

## Contribution

It provides a detailed analysis of path weights in undirected graphical models, including their interpretation and extension to various association measures.

## Key findings

- Path weights can be factorised into partial covariance and inflation factors.
- Covariance decomposition can be extended to correlation and inflated correlation.
- Application demonstrates usefulness in dietary intake network analysis.

## Abstract

A graphical model provides a compact and efficient representation of the association structure of a multivariate distribution by means of a graph. Relevant features of the distribution are represented by vertices, edges and other higher-order graphical structures, such as cliques or paths. Typically, paths play a central role in these models because they determine the independence relationships among variables. However, while a theory of path coefficients is available in models for directed graphs, little has been investigated about the strength of the association represented by a path in an undirected graph. Essentially, it has been shown that the covariance between two variables can be decomposed into a sum of weights associated with each of the paths connecting the two variables in the corresponding concentration graph. In this context, we consider concentration graph models and provide an extensive analysis of the properties of path weights and their interpretation. More specifically, we give an interpretation of covariance weights through their factorisation into a partial covariance and an inflation factor. We then extend the covariance decomposition over the paths of an undirected graph to other measures of association, such as the marginal correlation coefficient and a quantity that we call the inflated correlation. The usefulness of these findings is illustrated with an application to the analysis of dietary intake networks.