In recent years, there has been a growing need to understand large networks and to devise effective strategies to analyze them. In this dissertation, our main objectives are to understand various structural properties of large networks under suitable general framework and develop general techniques to analyze important network models arising from applied fields of study.

In the first part of this dissertation, we investigate properties of large networks that satisfy certain local conditions. In particular, we show that if the number of neighbors of each vertex and co-neighbors of each pair of vertices satises certain conditions then the number of copies of moderately large subgraphs is approximately same as that of an Erdos-Renyi random graph with appropriate edge density. We apply our results to different graph ensembles including exponential random graph models (ERGMs), thresholded graphs from high-dimensional correlation networks, Erdos-Renyi random graphs conditioned on large cliques and random d-regular graphs.

In the second part of this dissertation, we study models of weighted exponential random graphs in the large network limit. These models have recently been proposed to model weighted network data arising from a host of applications including socio-econometric data such as migration flows and neuroscience. We derive limiting results for the structure of these models as the number of nodes goes to infinity. We also derive sucient conditions for continuity of functionals in the specification of the model including conditions on nodal covariates.

Finally, we study site percolation on a class of non-regular graphs satisfying some mild assumptions on the number of neighbors of each vertex and co-neighbors of each pair of vertices. We show that there is a sharp phase transition (in site percolation) for the class of graphs under consideration and that in the supercritical regime the giant component is unique.