With respect to the available statistical modeling cross-sectional network data one may roughly distinguish between two strands: (a) models which explain the existence of an edge depends on nodal random effects, (b) models where the existence of an edge also depends on the local network structure. The strand (a) is phrased as p1 and p2 models and the strand (b) is based on exponential random graph models (ERGMs). We present a comprehensive inferential framework for Bayesian ERGMs with nodal random effects in order to account for both global dependence structure and network heterogeneity. Parameter inference and model selection procedures are based on the use of an approximate exchange algorithm and its trans-dimensional extension.

**Keywords:** social network analysis, network heterogeneity, exponential random graphs, exchange algorithm.

*Join work with Stephanie Thiemichen (LMU Munich), Goeran Kauermann (LMU Munich) and Nial Friel (UCD Dublin)