Local Information for Global Network Estimation in Latent Space Models
In many social networks, an individual observes only a restricted local view of the full network structure. We study such local views under a partial information framework that models an individual's observations as a subgraph based on path length, and address the problem of estimating a general latent space model from a single individual's local view. Compared to the full network, the partial information network contains many missing edges and depends on a random, potentially sparse neighborhood, posing significant challenges for estimation. We propose a projected gradient descent algorithm for maximum likelihood estimation and establish theoretical guarantees for its convergence under both conditional likelihood and full likelihood settings. To characterize the quality of a local view, we introduce an imbalance measure as a theoretical and diagnostic quantity for assessing bias in a local view and show that it plays a central role in determining convergence rates and estimation error bounds. Using simulated networks, we demonstrate that satisfactory estimation is possible from a single local view. In an application to U.S. Congress cosponsorship networks, we show how the estimated latent positions reveal nuanced structure in legislators' social relationships.