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

TL;DR

This paper identifies and analyzes two distinct sources of nonidentifiability in latent position random graph models, providing theoretical characterizations and limits for each type in various graph settings.

Contribution

It introduces and differentiates subspace and model-based nonidentifiability, offering new theoretical insights and limit results in random graph inference.

Findings

Characterization of subspace and model-based nonidentifiability

Limit results for model-based nonidentifiability with and without subspace nonidentifiability

Asymptotic behavior of covariances and U-statistics in stochastic block models

Abstract

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.