How many asymmetric communities are there in multi-layer directed networks?

Multilayer networks allow the modeling of interactions between the same entities across different contexts, such as temporal observations, varying settings, or interactions of different types. The goal of community detection in multilayer networks is to identify groups of nodes exhibiting similar connectivity patterns, which may vary across layers. We propose a method based on a joint nonnegative symmetric matrix trifactorization for community detection in multilayer networks, where each graph is approximated by a nonnegative symmetric matrix trifactorization. Our approach enforces constraints on the factor matrices so that communities are disjoint and shared across layers, while allowing each layer to have its own connectivity patterns and node degrees. This flexibility enables the model to capture both local and global structural variations across layers. We also develop an algorithm to efficiently solve this problem. We evaluate multilayer community detection methods using the multilayer degree-corrected stochastic block model (MDCBM), a flexible framework for generating realistic multilayer graphs with heterogeneous degrees and varying connectivity patterns. Experiments show that our method reliably detects communities across diverse regimes, whereas existing state-of-the-art approaches are often limited by restrictive structural assumptions.

We study clustering for multi-layer networks, where each observation is an undirected network measured on the same set of vertices. Under the mixture multi-layer stochastic block model (MMSBM), each network is generated from one of two stochastic block models, and the goal is to recover the latent network labels. We establish the minimax optimal network clustering error rate, which takes an exponential form and is characterized by the Chernoff information between the edge probability distributions of the component networks. We propose a novel two-stage estimator that combines tensor-based initialization, node and sample splitting, and a likelihood-based refinement. The refinement step exploits estimated node communities and achieves the minimax rate with high probability under extreme network sparsity. We further extend the likelihood-refinement framework to count-weighted networks through a mixture multi-layer Poisson block model. Numerical studies and two real-data analyses illustrate the finite-sample behavior of the proposed method and its use for both external-label benchmarking and unlabeled exploratory network clustering.

We study community recovery in the sparse symmetric stochastic block model with $q$ communities, average degree $d$ and signal strength $λ$ through statistical decision theory and Fisher information, and obtain three characterizations of the Kesten-Stigum threshold $dλ^2=1$ and of the information-computation gap below it. First, on each community-size profile the minimax risk of any class of rules closed under averaging and vertex relabeling equals its Bayes risk under the uniform prior; the posterior mean is the unique Bayes rule and is admissible, and the Bayes risk of degree-$D$ polynomial rules is the trivial risk times $1-\mathrm{Corr}_D^2$. Combined with known low-degree and information-theoretic results, this gives the gap as a worst-case statement: for $q\ge 5$ there is a window below the threshold in which no low-degree rule beats the trivial risk asymptotically, while an exponential-time rule does on a set of labelings of probability $1-o(1)$. Second, the Fisher information about $λ$ carried by cycle counts is a series with terms of order $k(dλ^2)^k$, convergent exactly when $dλ^2<1$; below the threshold the relative error of every unbiased cycle-based estimator of $λ^k$ stays above an explicit constant, and every cycle-count test has success probability bounded below one. Third, the derivative of belief propagation at its uninformative fixed point multiplies a random perturbation by $|λ|\sqrt{d}$ per iteration, and one EM step taken there leaves $λ$ unchanged. A signal-to-noise computation recovers the condition $dλ^{1/χ}>1$ of Chin et al. for $q=n^χ$ communities and identifies personalized PageRank as a walk count with suboptimal weights. Experiments on networks with up to $3\times 10^5$ vertices confirm the threshold for $q=2$, the hard window for $q=5$, and the many-community scaling.