Maximum Likelihood Estimation for Network Models with Latent Geometry under Snowball Sampling
Snowball sampling is a widely used design for collecting network data from large or hard-to-reach populations, yet naive inference that ignores the sampling mechanism produces systematically biased parameter estimates. We derive the exact likelihood of a multi-wave snowball sample for the class of continuous latent space (CLS) models, in which edges form independently conditional on latent vertex-level quantities, and show that conditional edge independence reduces the marginalization over unobserved network configurations to a closed-form expression portable across the entire CLS class. We develop a stochastic Expectation-Maximization algorithm for the Euclidean latent distance model as a concrete implementation, and apply the framework to the large-scale co-inventor network of German semiconductor patent applicants by drawing multiple snowball samples. We find that the naive procedure severely underestimates latent space variance, produces networks with nearly twice the observed edge count, and achieves a spectral goodness-of-fit nine times worse than the corrected model, which directly affects the quantitative interpretation of covariate effects.