This paper develops new limit theory for data that are generated by networks or more generally display cross-sectional dependence structures that are governed by observable and unobservable characteristics. Strategic network formation models are an example. Wether two data points are highly correlated or not depends on draws from underlying characteristics distributions. The paper defines a measure of closeness that depends on primitive conditions on the distribution of observable characteristics as well as functional form of the underlying model. A summability condition over the probability distribution of observable characteristics is shown to be a critical ingredient in establishing limit results. The paper establishes weak and strong laws of large numbers as well as a stable central limit theorem for a class of statistics that include as special cases network statistics such as average node degrees or average peer characteristics. Some worked examples illustrating the theory are provided.