The Geometry of Community Detection via the MMSE Matrix

Mutual Information in Community Detection with Covariate Information and Correlated Networks

“…Without loss of generality these labels can be embedded into finite dimensional Euclidean space. To facilitate the exposition of our results, we use the whitened representation described in [13], where the labels are supported on a set of k points in {µ 1 , . .…”

“…there is an edge between nodes i and j and zero otherwise. Following [13], each network is drawn according to a degree-balanced SBM of the form…”

“…The information-theoretic limits describe the optimal performance that can be attained without any constraints on computational complexity. For network models with k ≥ 4 communities [2,5,20] or asymmetries [5,13], there exists a computational-to-statistical gap, where weak recovery below the Kesten-Stigum threshold is information-theoretically possible, even though all known polynomial-time algorithms fail in this regime.…”

“…General SBMs have been considered by Abbe and Sandon [10] who characterize conditions for weak recovery, Lesieur et al [7] who analyze the performance of an approximate message passing algorithm, and Reeves et. al [13] who study the asymptotic per-node mutual information and MMSE in degree-balanced SBMs.…”

Hamilton-Jacobi equations for inference of matrix tensor products

“…A generalization of these spiked matrix models can be seen in the study of community detection problems and the stochastic block models. In certain settings, the community detection problem is asymptotically equivalent to Y = 2t N XBX ⊺ + W where B is deterministic and models the community interactions (see [30,31]). More generally, the community detection with several correlated networks is asymptotically equivalent to the multiview spiked matrix model Y l = 2t N XB l X ⊺ + W l for l = 1, 2, .…”