Asymptotic uncertainty quantification for communities in sparse planted bi-section models

TL;DR

This paper investigates the asymptotic behavior of posterior distributions for community detection in sparse planted bi-section models, establishing conditions for exact recovery and confidence set validity in various sparsity regimes.

Contribution

It provides sharp bounds for recovery regimes and links credible sets to frequentist confidence sets under different sparsity conditions.

Findings

Exact recovery in regimes with $O(rac{ ext{log}(n)}{n})$ edge probabilities.

Credible sets can serve as asymptotic confidence sets under certain conditions.

Enlarged credible sets achieve frequentist coverage in very sparse regimes.

Abstract

Posterior distributions for community structure in sparse planted bi-section models are shown to achieve exact (resp. almost-exact) recovery, with sharp bounds for the sparsity regimes where edge probabilities decrease as $O(\log(n)/n)$ (resp. $O(1/n)$). Assuming posterior recovery, one may interpret credible sets (resp. enlarged credible sets) as asymptotically consistent confidence sets; the diameters of those credible sets are controlled by the rate of posterior concentration. If credible levels are chosen to grow to one quickly enough, corresponding credible sets can be interpreted as frequentist confidence sets without conditions on posterior concentration. In the regimes with $O(1/n)$ edge sparsity, or when within-community and between-community edge probabilities are very close, credible sets may be enlarged to achieve frequentist asymptotic coverage, also without conditions on posterior concentration.