Conditional estimation of exponential random graph models from snowball sampling designs

• Snowball sampling designs for networks lead to partial observations on network ties.
• Exponential random graph models (ERGM) are a general class of models for networks.
• We propose a conditional estimation approach for ERGM parameters from a snowball sample.
• We demonstrate via simulation the effectiveness of the conditional estimation method.

A complete survey of a network in a large population may be prohibitively difficult and costly. So it is important to estimate models for networks using data from various network sampling designs, such as link-tracing designs. We focus here on snowball sampling designs, designs in which the members of an initial sample of network members are asked to nominate their network partners, their network partners are then traced and asked to nominate their network partners, and so on. We assume an exponential random graph model (ERGM) of a particular parametric form and outline a conditional maximum likelihood estimation procedure for obtaining estimates of ERGM parameters. This procedure is intended to complement the likelihood approach developed by  Handcock and Gile (2010) by providing a practical means of estimation when the size of the complete network is unknown and/or the complete network is very large. We report the outcome of a simulation study with a known model designed to assess the impact of initial sample size, population size, and number of sampling waves on properties of the estimates. We conclude with a discussion of the potential applications and further developments of the approach.