Graphical Models Over Heterogeneous Domains and for Multilevel Networks

We review models for analyzing multivariate data of mixed (heterogeneous) domains such as binary, categorical, ordinal, counts, continuous, and/or skewed continuous, and methods for modeling various graphs including multiplex, multilevel, and multilayer networks. Data are modeled with Markov random fields which encode Markov property between nodes: two nodes are not connected with an edge if and only if random variables associated with these nodes are conditionally independent, given the other variables. Inferring dependence structure through graphical models (both directed and undirected) is essential for discovering multivariate interaction among high-dimensional data, which could potentially be associated with several diseases. Networks are modeled with exponential random graph models which encode Markov property between edges: two edges are conditionally dependent, given the rest of the network, if they have a common vertex. Studying and understanding multilayer and/or multilevel representations of various phenomena, including social and natural phenomena, could lead to predictive models of these phenomena. Modeling data of heterogeneous domains and multilevel and/or multilayer networks pose challenges which are reviewed. Addressing these challenges within a unified framework stresses open problems and points out new directions for research.