Normal Approximation in Large Network Models

Abstract

We prove central limit theorems for models of network formation and network processes with homophilous agents. The results hold under large-network asymptotics, enabling inference in the typical setting where the sample consists of a small set of large networks. We first establish a general central limit theorem under high-level "stabilization" conditions that provide a useful formulation of weak dependence, particularly in models with strategic interactions. The result delivers a ? n rate of convergence and a closed-form expression for the asymptotic variance. Then using techniques in branching process theory, we derive primitive conditions for stabilization in the following applications: static and dynamic models of strategic network formation, network regressions, and treatment effects with network spillovers. Finally, we suggest some practical methods for inference.JEL Codes: C31, C57, D85Network models have attracted considerable attention in economics as tractable representations of non-market interactions, such as peer effects and social learning, and formal economic relations, such as financial and trade networks. However, for many important network models, methods for inference are unavailable due to the lack of a large-sample theory. The novel feature of network data is that it typically takes the form of observations on a small set of large networks. Additionally, many network models of interest feature strategic interactions or spillovers, which generate dependence between network subunits. A large-sample theory must outline conditions under which the amount of "independent information" grows with the number of nodes or agents in the network, despite network autocorrelation.The main contribution of this paper is a large-network central limit theorem (CLT) applicable to a large class of network moments. These moments can be written as averages of node-level statistics, namelywhere X i is a vector of homophilous attributes for node i, 1 X n the set of all nodes' homophilous attributes, W the set of all other node attributes, and A the observed network or network time series on n nodes. A simple example is the degree of node i, ψpX i , X n , W, Aq " ř j‰i A ij , which is the number of links involving i. More generally, these moments may be complicated functionals of the network, including the average clustering coefficient, subnetwork counts, and regression estimators.There are two main technical contributions of the paper. First, we derive a new CLT that holds under high-level "stabilization" conditions on the node statistic ψpX i , X n , W, Aq. These provide a general formulation of network weak dependence for which we can derive primitive conditions in a variety of models. The main requirement is that i's node statistic only depends on a random "relevant set" of nodes with asymptotically bounded size that effectively constitutes i's dependency neighborhood. Our conditions are modifications of assumptions used in the stochastic geometry literature (e.g. Penrose and Yukich, 2001; Penrose, 2003), which f...

Equilibrium computation in discrete network games

“…Leung (2019) used this condition to prove a CLT for graphical games. An analogous condition is used by Leung and Moon (2019) to obtain a CLT for network formation games. The main theoretical contribution of this paper is to show that this condition also enables feasible computation of .…”

“…We approximate this by a branching process for which the desired tail bounds can be obtained more easily, which is a common technique in random graph theory (Bollobás and Riordan (2008)). The argument is also used in Leung and Moon (2019) and Leung (2019) to derive primitive conditions for CLTs in network formation games and graphical games, respectively. We generalize the tail bounds of the former paper to a larger class of sparse graphs.…”

Who should get vaccinated? Individualized allocation of vaccines over SIR network

“…We also note that there is a growing literature on estimation of treatment effects under network interference. Manski (2013) Basse et al (2019), and Leung and Moon (2019). Li and Wager (2020) non-parametrically estimates direct and indirect effects of treatment in a random network setting.…”

Network Data

“…This mechanism generates sufficient independence among distant units such that LLNs and CLTs can be proven. Leung (2019), Kuersteiner (2019), and Leung & Moon (2019) develop these ideas to prove LLNs and CLTs for network statistics where the observed network is assumed to be a strategic network equilibrium configuration. A challenge of this approach is that valid inference appears to require information on agents' positions (so that HAC type variance estimators can be used).…”