A two-stage working model strategy for network analysis under hierarchical exponential random graph models

Social network data are complex and dependent data. At the macro-level, social networks often exhibit clustering in the sense that social networks consist of communities; and at the micro-level, social networks often exhibit complex network features such as transitivity within communities. Modeling real-world social networks requires modeling both the macro- and micro-level, but many existing models focus on one of them while neglecting the other. In recent work, [28] introduced a class of Exponential Random Graph Models (ERGMs) capturing community structure as well as microlevel features within communities. While attractive, existing approaches to estimating ERGMs with community structure are not scalable. We propose here a scalable two-stage strategy to estimate an important class of ERGMs with community structure, which induces transitivity within communities. At the first stage, we use an approximate model, called working model, to estimate the community structure. At the second stage, we use ERGMs with geometrically weighted dyadwise and edgewise shared partner terms to capture refined forms of transitivity within communities. We use simulations to demonstrate the performance of the two-stage strategy in terms of the estimated community structure. In addition, we show that the estimated ERGMs with geometrically weighted dyadwise and edgewise shared partner terms within communities outperform the working model in terms of goodness-of-fit. Last, but not least, we present an application to high-resolution human contact network data.