Abstract
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.
Highlights
Graphical models provide a compact and efficient representation of the association structure of a multivariate distribution by means of a graph and have become a popular tool for representing network structures in many applied contexts; see Maathuis et al (2019) for a recent review of the state of art of graphical models
The path xy = ⟨x, 1, 2, y⟩ has inflated correlation weight equal to = 0.09 and if we apply Proposition 5 with respect to the natural vertex numbering starting from the endpoint x we can associate to every vertex of the path an inflation factor and to every edge a partial correlation, as follows, where we write uv∣rest to denote the partial correlation between Xu and Xv given all the remaining variables XV⧵{u,v}
Paths play a central role in undirected graphical models and are the key structures to be used in the identification, for instance, of relevant patterns and of vertex which may be regarded as central
Summary
Graphical models provide a compact and efficient representation of the association structure of a multivariate distribution by means of a graph and have become a popular tool for representing network structures in many applied contexts; see Maathuis et al (2019) for a recent review of the state of art of graphical models. If XV is a vector of continuous random variables an undirected network, called a concentration graph of XV , is constructed in such a way that every vertex is associated with a variable and a missing edge between two vertices implies that the corresponding partial correlation is equal to zero (Lauritzen 1996) In this way, the association structure of XV is encoded by the paths connecting the variables. Every edge is associated with a partial correlation quantifying the contribution to the path of the corresponding pairwise association 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 building blocks forming the path.
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