Modeling and monitoring multilayer attributed weighted directed networks via a generative model

As data with network structures are widely seen in diverse applications, the modeling and monitoring of network data have drawn considerable attention in recent years. When individuals in a network have multiple types of interactions, a multilayer network model should be considered to better characterize its behavior. Most existing network models have concentrated on characterizing the topological structure among individuals, and important attributes of individuals are largely disregarded in existing works. In this article, first, we propose a unified static Network Generative Model (static-NGM), which incorporates individual attributes in network topology modeling. The proposed model can be utilized for a general multilayer network with weighted and directed edges. A variational expectation maximization algorithm is developed to estimate model parameters. Second, to characterize the time-dependent property of a network sequence and perform network monitoring, we extend the static-NGM model to a sequential version, namely, the sequential-NGM model, with the Markov assumption. Last, a sequential-NGM chart is developed to detect shifts and identify root causes of shifts in a network sequence. Extensive simulation experiments show that considering attributes improves the parameter estimation accuracy and that the proposed monitoring method also outperforms the three competitive approaches, static-NGM chart, score test-based chart (ST chart) and Bayes factor-based chart (BF chart), in both shift detection and root cause diagnosis. We also perform a case study with Enron E-mail data; the results further validate the proposed method.