A Random-Field Approach to Inference in Large Models of Network Formation

Abstract. We establish a law of large numbers and central limit theorem for a large class of network statistics, enabling inference in models of network formation with only a single network observation. Our model allows the decision of an individual (node) to form associations (links) to depend quite generally on the endogenous structure of the network. Formally, we prove that certain node-level functions of the network constitute α-mixing random fields, objects for which central limit theorems exist. The key assumptions are that (1) nodes endowed with similar attributes prefer to link (homophily); (2) there is enough “diversity” in node attributes; and (3) certain latent sets of nodes that are unconnected form their links independently (“isolated societies” do not “coordinate”). Our results enable the estimation of certain network moments that are useful for inference. We leverage these moments to construct moment inequalities that define bounds on the identified set. Relative to feasible alternatives, these bounds are sharper and computationally tractable under weaker restrictions on network externalities.