This thesis aims to study graph networks of similar structure and statistical distribution and make inferences based on the nodal and dyadic covariates. Incorporating a small amount of randomness can drastically alter important features of the underlying structure and dynamics of a network. Our principal goal is to estimate model parameters from a given network, generate random graphs of similar structure with these models, and evaluate how adequately the model represents the observed network. In addition, we also measure edge importance through model deviance. Given a graph network, you essentially have a sample of size one from an unknown population of networks. To evaluate our results, we use the “Spectral Goodness-of-Fit” (SGOF) statistic Shore and Lubin (2015) based on eigenvalues of the graph Laplacian to quantify goodness-of-fit between our observed graph and graph simulated from fitted models. We then adapt a similar GOF approach to other graph measures, examining goodness of fit to labelled graph properties. We finally describe two recent applications of random graph models. For the purposes of this thesis, we consider simple undirected graph networks: graphs in which the edges have no orientation; with no multiple edges and no self-loops in the graph.