We develop a new class of random graph models for the statistical estimation of network formation -- subgraph generated models (SUGMs). Various subgraphs -- e.g., links, triangles, cliques, stars -- are generated and their union results in a network. We show that SUGMs are identified and establish the consistency and asymptotic distribution of parameter estimators in empirically relevant cases. We show that a simple four-parameter SUGM matches basic patterns in empirical networks more closely than four standard models (with many more dimensions): (i) stochastic block models; (ii) models with node-level unobserved heterogeneity; (iii) latent space models; (iv) exponential random graphs. We illustrate the framework's value via several applications using networks from rural India. We study whether network structure helps enforce risk-sharing and whether cross-caste interactions are more likely to be private. We also develop a new central limit theorem for correlated random variables, which is required to prove our results and is of independent interest.