Convergence-rate behavior at parameter-space boundaries

Determine the asymptotic convergence rate of the Graph Pencil Method—i.e., the estimator that maps a finite set of star and bistar subgraph densities to the parameters (π,B) of a degree-separated K-block stochastic block model—when the true connection probabilities lie on or near the boundary of the parameter space, specifically when some entries of B approach 0 or 1. Characterize how this boundary behavior differs from the interior regime where the method is known to be well-behaved.

Background

The paper introduces the Graph Pencil Method, which uses subgraph densities (notably stars and bistars) to recover parameters of degree-separated stochastic block models with K blocks. Empirically, the method shows strong performance and appears to attain optimal rates in interior regimes where parameters are well within the allowable domain.

However, the authors note a lack of theoretical understanding for behavior at the boundaries of the parameter space, such as when some block-to-block connection probabilities are very close to 0 or 1. They point out that, near these boundaries, the method can in principle output invalid probabilities due to the absence of explicit constraints enforcing values within [0,1], highlighting the need to precisely characterize convergence rates in these edge cases.