On Two Distinct Sources of Nonidentifiability in Latent Position Random Graph Models
This work addresses theoretical issues in random graph inference, which is incremental as it clarifies existing nonidentifiability problems without introducing new methods.
The paper tackles the problem of nonidentifiability in latent position random graph models by defining and analyzing two distinct sources, subspace and model-based nonidentifiability, and characterizing their limits, including results for stochastic block models and generalized random dot product graphs.
Two separate and distinct sources of nonidentifiability arise naturally in the context of latent position random graph models, though neither are unique to this setting. In this paper we define and examine these two nonidentifiabilities, dubbed subspace nonidentifiability and model-based nonidentifiability, in the context of random graph inference. We give examples where each type of nonidentifiability comes into play, and we show how in certain settings one need worry about one or the other type of nonidentifiability. Then, we characterize the limit for model-based nonidentifiability both with and without subspace nonidentifiability. We further obtain additional limiting results for covariances and $U$-statistics of stochastic block models and generalized random dot product graphs.