Finite-sample calibration of the Parisi-based detection threshold δ_n

Ascertain a principled choice of the threshold δ_n in the decision rule T(G) that compares the ground state energy _n(A_G; Σ_n) to the Parisi value for detecting a two-community structure in the stochastic block model, with guarantees on type-I and type-II error probabilities at finite n.

Background

The paper proposes a detection test based on comparing the observed ground state energy of the minimum bisection objective to the Parisi limit, but notes that theory only ensures that any positive constant works asymptotically without prescribing a finite-n threshold.

This leaves a statistical calibration problem: selecting δ_n to control error rates for practical finite-size graphs, beyond asymptotic guarantees.

References

Of course, this approach runs into two difficulties: $(i)$~We do not know how to set $\delta_n$ (from a statistics perspective, Theorem \ref{thm:parisi} merely says that any positive constant will work for $n$ large enough); $(ii)$~In general, we do not know how to evaluate $\mathsf{OPT}_n(A;\Sigma_n)$. Some of the developments discussed in Section \ref{sec:Algo} address the last problem.

— Spin Glass Concepts in Computer Science, Statistics, and Learning  (2602.23326 - Montanari, 26 Feb 2026) in Section 2 (Parisi’s formula), final remark on detection using Parisi’s value