Dynamic factorial graphical models for dynamic networks

Dynamic networks models describe a growing number of important scientific processes, from cell biology and epidemiology to sociology and finance. Estimating dynamic networks from noisy time series data is a difficult task since the number of components involved in the system is very large. As a result, the number of parameters to be estimated is typically larger than the number of observations. However, a characteristic of many real life networks is that they are sparse. For example, the molec- ular structure of genes make interactions with other components a highly-structured and, therefore, a sparse process. Penalized Gaussian graphical models have been used to estimate sparse networks. However, the literature has focussed on static networks, which lack specific temporal interpretations. We propose a flexible collection of ANOVA-like dynamic network models, where the user can select specific time dynamics, known presence or absence of links and a particular autoregressive structure. We use undirected graphical models with block equality constraints on the parameters. This reduces the number of parameters, increases the accuracy of the estimates and makes interpretation of the results more relevant. We show that the constrained likelihood optimization problem can be solved by taking advantage of an efficient solver, LogdetPPA, developed in convex optimization. Model selection strategies can be used to select a particular model. We illustrate the flexibility of the method on both synthetic and real data.