Path-dependent parametric decompositions in Ising models

The analysis of paths in undirected graph models can be used to quantify the relevance of the strength of association in multiple paths connecting a pair of vertices of the graph. Some results are available in multivariate Gaussian settings as the covariance of two variables can be decomposed into the sum of measures related to paths joining the variables of the underlying graph. This paper studies the analysis of paths in undirected graph models for binary data, with special focus on Ising models, where the propagation of the variable status through multiple paths joining a pair of vertices is an aspect of interest. A novel logistic regression approach for baseline events in multi-way tables is proposed to show that a parameter of pairwise association can be computed by the sum of components related to paths. These components are based on products of odds ratios which are typically used to measure the dependence represented by the edges in Ising models. Specifically, two parametric decompositions are developed to gain insight on a twofold aspect of interest: the relevance of the multivariate dependence within each path connecting a pair of vertices and the interaction between the multivariate dependence in each path and in the rest of the graph. The results are illustrated through an application to cyber-security risk assessment in industrial networks.