Exponential-Family Random Graph Models with Time Varying Network Parameters

Dynamic networks are a general language for describing time-evolving complex systems, and have long been an interesting research area. It is a fundamental research question to model time varying network parameters. However, due to difficulties in modeling functional network parameters, there is little progress in the current literature to effectively model time varying network parameters. In this work, we consider the situation in which network parameters are univariate nonparametric functions instead of constants. Using a kernel regression techniques, we introduce a novel unified procedure to effectively estimate those functional network parameters in the exponential-family random graph models. Moreover, by adopting the finite mixture models, we extend our model to mixture of exponential-family random graph models with functional network parameters which simultaneously allows both modeling and detecting communities for the dynamic networks. To choose optimal number of communities and kernel bandwidth, we propose conditional likelihood BIC and choose bandwidth by adopting the idea of network cross validation. Furthermore, we design an efficient variational expectationmaximization algorithm to find approximate maximum local likelihood estimates of network parameters and global estimates of mixing proportions. The power of our method is demonstrated in depth simulation studies and real-world applications to dynamic arm trade networks.