We will address three aspects of statistical methodology for Exponential family Random Graph Models (ERGMs) in the context of applications to social network analysis. We start by addressing the topic of degeneracy in ERGMs. This is a problem often misunderstood to characterize the entire family of ERGMs, but is properly understood as a more limited issue of model misspecification. We will derive our intuition from a classic degenerate specification, develop some new diagnostic tools that exploit the geometry of ERGMs, and provide general guidance on alternative specifications that can reduce degeneracy. In the second chapter we will address the methodology of estimating partnership duration in the context of potential social network dependence. Partnership duration may be influenced by the attributes of the two persons involved (nodes), the attributes of the relationship itself (link), and the presence or absence of other links to these nodes in the network. In the latter case, there is dependence in durations across the links that make the problem incompatible with the assumptions of traditional survival analysis. We will develop a statistical framework in which the influence of these potential covariates and models can be compared and evaluated. For the last chapter, we will extend the recent advances in temporal ERGMs for co-evolution (or â€œselection-influenceâ€) models, in which both the edges and the vertex attributes are treated as random variables. We will develop the theory and methods for estimation of co-evolution models with longitudinal network data.