Spectral Recovery in the Labeled SBM

We consider the problem of exact community recovery in the Labeled Stochastic Block Model (LSBM) with $k$ communities, where each pair of vertices is associated with a label from the set $\{0,1, \dots, L\}$. A pair of vertices from communities $i,j$ is given label $\ell$ with probability $p_{ij}^{(\ell)}$, and the goal is to recover the community partition. We propose a simple spectral algorithm for exact community recovery, and show that it achieves the information-theoretic threshold in the logarithmic-degree regime, under the assumption that the eigenvalues of certain parameter matrices are distinct and nonzero. Our results generalize recent work of Dhara, Gaudio, Mossel, and Sandon (2023), who showed that a spectral algorithm achieves the information-theoretic threshold in the Censored SBM, which is equivalent to the LSBM with $L = 2$. Interestingly, their algorithm uses eigenvectors from two matrix representations of the graph, while our algorithm uses eigenvectors from $L$ matrices.