Latent Agents in Networks: Estimation and Targeting

We consider a platform that serves (observable) agents, who belong to a larger network that also includes additional agents who are not served by the platform. We refer to the latter group of agents as latent agents. Associated with each agent are the agent's covariate and outcome. The platform has access to past covariates and outcomes of the observable agents, but no data for the latent agents is available to the platform. Crucially, the agents influence each other's outcome through a certain influence structure. In particular, observable agents influence each other both directly and indirectly through the influence they exert on the latent agents. The platform doesn't know the inference structure of either the observable or the latent parts of the network. We investigate how the platform can estimate the dependence of the observable agents' outcomes on their covariates, taking the presence of the latent agents into account. First, we show that a certain matrix succinctly captures the relationship between the outcomes and the covariates. We provide an algorithm that estimates this matrix using historical data of covariates and outcomes for the observable agents under a suitable approximate sparsity condition. We also establish convergence rates for the proposed estimator despite the high dimensionality that allows more agents than observations. Second, we show that the approximate sparsity condition holds under the standard conditions used in the literature. Hence, our results apply to a large class of networks. Finally, we illustrate the applications to a targeted advertising problem. We show that, by using the available historical data with our estimator, it is possible to obtain asymptotically optimal advertising decisions despite the presence of latent agents.