On Two Distinct Sources of Nonidentifiability in Latent Position Random Graph Models

The statistical analysis of network data is important for fields such as neuroscience (Vogelstein et al., 2012), sociology (Hoff et al., 2002), and physics (Newman and Girvan, 2004; Bickel and Chen, 2009). Recently, network data have become ubiquitous in the the modern data-science landscape, and a large literature on statistical methods for analyzing these data has developed. Popular statistical models for conditionally independent random graphs include, but are not limited to, the stochastic block model (Holland et al., 1983), the random dot product graph (Young and Scheinerman, 2007; Athreya et al., 2017), and graphons (Lovász, 2012; Diaconis and Janson, 2007). Both the stochastic block model and the random dot product graph are examples of latent position random graphs (Hoff et al., 2002), a graph model that is motivated by the idea that individual nodes have latent positions whose values determine their propensity to form connections. The purpose of this manuscript is to explain a curious phenomenon that arises in latent position random graph settings.