Path weights in concentration graphs

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 finding is illustrated with an application to the analysis of dietary intake networks.

On the interpretation of inflated correlation path weights in concentration graphs

“…Jones and West (2005) considered the measure of association between two variables provided by the covariance and showed that this quantity can be decomposed in terms of additive weights associated with the paths joining the corresponding vertices. Roverato and Castelo (2020) provided an analysis of the properties of the covariance path weights introduced by Jones and West (2005) and showed that inflation factors play a key role in the interpretation of these quantities; see also Castelo (2017, 2018); Peeters et al (2020).…”

“…The comparison of paths with different endpoint requires the use of normalized measures of association and, to this aim, Roverato and Castelo (2020) considered the weights obtained for the decomposition of correlation coefficients. Furthermore, they introduced a novel normalized measure of linear association, named the inflated correlation coefficient, and showed that the weights obtained from the decomposition of this quantity satisfy useful properties that, as far as the strength of paths is of concern, make them an appealing alternative to the classical correlation coefficients.…”

“…Hence, in order to compare paths with different endpoints it is necessary to deal with normalized quantities. Roverato and Castelo (2020) noticed that the decomposition in (3) is not restricted to covariance matrices but it can be straightforwardly extended to any positive definite matrix = { uv } u,v∈V obtained as = where = { uv } u,v∈V is a diagonal matrix with nonzero diagonal entries. More specifically, both and V are specific instances of this general setting with = diag( ) − 1 2 and = diag(K) 1 2 , respectively.…”

“…Indeed, both correlations and inflated correlations are normalized measures of association and for this reason the corresponding weights are of interest. More specifically, Roverato and Castelo (2020) provided the following decomposition of inflated correlations, where we refer to Roverato and Castelo (2020) for details on the properties of inflated correlation weights. Here we remark that, as well as the weight in (4) also inflated correlation weights can be factorized into a partial weight and an inflation factor, 𝜔(𝜋, 𝛺 V ) = 𝜔(𝜋, 𝛺 V PP⋅ P) × IF P and, furthermore, all the factors in (6) are feasible of a clear interpretation because it is the product of the partial correlations corresponding to the edges of the path and of | V PP | ≥ 1 which "inflates" the partial correlations.…”

“…Although centrality is most commonly computed for vertices, also edge centrality is of interest; see Girvan and Newman (2002); Bröhl and Lehnertz (2019) and references therein. Hence, similarly to (18), we compute betweenness of an edge {u, v} according to the criterion , as Some applications of betweenness centrality in graphical models can be found in Bringmann et al ( 2019), Dablander and Hinne (2019), Peeters et al (2020) and Roverato and Castelo (2020). However, the construction of centrality measures specifically designed to suit the graphical model framework is a recent, and largely unexplored, area of research.…”