Network‐Valued Random Vectors and Their Statistical Foundations: A Comparison of Inferential Techniques and Testing Frameworks Under Heterogeneity

ABSTRACT

Network‐valued random vectors (NVRVs) provide a statistical framework for settings in which each observational unit is a network rather than a scalar, vector, image, or functional observation. Such data occur in social networks, omics and gene‐regulatory systems, functional brain connectivity, policy and intervention networks, and other domains where relational structure is itself the object of inference. NVRVs incorporate dependence through structured interactions among nodes and edges, thereby presenting significant challenges for statistical modeling, regression, and inference. This article provides an advanced review of inference techniques for NVRVs, with technical expositions focusing on the problem of measuring and testing network change in dynamic and heterogeneous settings. We review approaches for modeling networks as both responses and covariates, emphasizing key strategies such as edge‐wise models, summary‐based regression, latent variable methods, and Bayesian hierarchical formulations. We discuss how heterogeneity and temporal dynamics complicate inference, particularly in the context of network change detection, where current approaches frequently target isolated network features instead of the full network distribution. We consider the perspectives from both frequentist and Bayesian paradigms to identify fundamental gaps in current methodology including limitations in global network testing and the lack of theoretical guarantees in heterogeneous network data. Using the MRN‐114 dataset from the Mind Research Network database as empirical test cases, we perform integrative experiments demonstrating use of the techniques, discussing the advantages and pitfalls of each. Overall, our reviews compare techniques covering the representation of network‐valued observations, the measurement of dynamic change, and the inferential consequence of heterogeneity.