Online monitoring of directed count-weighted network with attributes via the gravity model

The emerging field of network science has witnessed significant growth with applications spanning transportation, social networks, and biological systems. As the volume of data generated in the form of network data streams continues to escalate, there is a growing need for advanced methodologies to monitor these complex networks. This article focuses on the development of a novel online monitoring framework for directed, count-weighted, and attributed networks. Our methodology incorporates the gravity model to elucidate the formation mechanisms inherent in directed and weighted networks with covariate information. By introducing directional node intensity parameters into a generalized linear model, we enhance the characterization of network edges, providing a more intuitive representation of both weight and direction. For online monitoring, we propose an exponentially weighted moving average (EWMA) control chart based on the weighted likelihood ratio test. This chart facilitates continuous online parameter estimation, offering a practical solution for monitoring evolving network structures. The effectiveness of the proposed methodologies is demonstrated through simulation studies and real-data applications, showing their applicability and advantages in diverse network scenarios.