  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.
