  In this paper, we explore the capability of both the Adjacency Spectral
Embedding (ASE) and the Graph Encoder Embedding (GEE) for capturing an embedded
pseudo-clique structure in the random dot product graph setting. In both theory
and experiments, we demonstrate that this pairing of model and methods can
yield worse results than the best existing spectral clique detection methods,
demonstrating at once the methods' potential inability to capture even modestly
sized pseudo-cliques and the methods' robustness to the model contamination
giving rise to the pseudo-clique structure. To further enrich our analysis, we
also consider the Variational Graph Auto-Encoder (VGAE) model in our simulation
and real data experiments.
