Weak recovery from Bethe–Hessian eigenvectors at all constant degrees

Establish that the informative eigenvectors of the Bethe–Hessian yield weak recovery in the stochastic block model throughout the full constant-expected-degree regime d>1.

Background

The paper proves that, for a stochastic block model with equal expected degree d>1, the number of negative eigenvalues of the Bethe–Hessian matches the number of informative planted directions. This result concerns the count of negative outliers, rather than the statistical usefulness of the associated eigenvectors or eigenspaces for community detection.

The authors note that weak recovery using Bethe–Hessian eigenvectors has been established for sufficiently large expected degree, but that their theorem does not prove this property throughout the entire constant-degree regime. They explicitly identify whether the corresponding eigenspaces yield weak recovery for every d>1 as remaining open.

References

Our theorem determines the number of negative outliers of the Bethe--Hessian, but not the corresponding eigenspaces. In particular, we do not prove that its eigenvectors yield weak recovery throughout the full constant-degree regime. This was proven for sufficiently large $d$ in , and, to our knowledge, remains open for general $d>1$.

— The Bethe-Hessian down to the Percolation Threshold  (2608.16672 - Dong et al., 17 Aug 2026) in Remark immediately following Theorem 1.1 in Section 2.3, “Our result”