Spectral Properties and Weak Detection in Stochastic Block Models

We consider the stochastic block model where connection between vertices is perturbed by some latent (and unobserved) random geometric graph. The objective is to prove that spectral methods are robust to this type of noise, even if they are agnostic to the presence (or not) of the random graph. We provide explicit regimes where the second eigenvector of the adjacency matrix is highly correlated to the true community vector (and therefore when weak/exact recovery is possible)....

Community detection is a fundamental problem in network science. In this paper, we consider community detection in hypergraphs drawn from the $hypergraph$ $stochastic$ $block$ $model$ (HSBM), with a focus on exact community recovery. We study the performance of polynomial-time algorithms which operate on the $similarity$ $matrix$ $W$, where $W_{ij}$ reports the number of hyperedges containing both $i$ and $j$. Under this information model, while the precise information-theore...

We derive rigorous bounds for well-defined community structure in complex networks for a stochastic block model (SBM) benchmark. In particular, we analyze the effect of inter-community "noise" (inter-community edges) on any "community detection" algorithm's ability to correctly group nodes assigned to a planted partition, a problem which has been proven to be NP complete in a standard rendition. Our result does not rely on the use of any one particular algorithm nor on the an...

Spectral clustering is a fast and popular algorithm for finding clusters in networks. Recently, Chaudhuri et al. (2012) and Amini et al.(2012) proposed inspired variations on the algorithm that artificially inflate the node degrees for improved statistical performance. The current paper extends the previous statistical estimation results to the more canonical spectral clustering algorithm in a way that removes any assumption on the minimum degree and provides guidance on the ...

In this paper, we analyse classical variants of the Spectral Clustering (SC) algorithm in the Dynamic Stochastic Block Model (DSBM). Existing results show that, in the relatively sparse case where the expected degree grows logarithmically with the number of nodes, guarantees in the static case can be extended to the dynamic case and yield improved error bounds when the DSBM is sufficiently smooth in time, that is, the communities do not change too much between two time steps....

Spectral clustering has been one of the widely used methods for community detection in networks. However, large-scale networks bring computational challenges to the eigenvalue decomposition therein. In this paper, we study the spectral clustering using randomized sketching algorithms from a statistical perspective, where we typically assume the network data are generated from a stochastic block model that is not necessarily of full rank. To do this, we first use the recently ...

The classical setting of community detection consists of networks exhibiting a clustered structure. To more accurately model real systems we consider a class of networks (i) whose edges may carry labels and (ii) which may lack a clustered structure. Specifically we assume that nodes possess latent attributes drawn from a general compact space and edges between two nodes are randomly generated and labeled according to some unknown distribution as a function of their latent att...

This paper provides a selective review of the statistical network analysis literature focused on clustering and inference problems for stochastic blockmodels and their variants. We survey asymptotic normality results for stochastic blockmodels as a means of thematically linking classical statistical concepts to contemporary research in network data analysis. Of note, multiple different forms of asymptotically Gaussian behavior arise in stochastic blockmodels and are useful fo...

In this paper, we prove the weak and strong consistency of the maximum and integrated conditional likelihood estimators for the community detection problem in the Stochastic Block Model with $k$ communities and unknown parameters. We show that maximum conditional likelihood achieves the optimal known threshold for exact recovery in the logarithmic degree regime. For the integrated conditional likelihood, we obtain a sub-optimal constant in the same regime. Both methods are sh...

The stochastic block model is a canonical model of communities in random graphs. It was introduced in the social sciences and statistics as a model of communities, and in theoretical computer science as an average case model for graph partitioning problems under the name of the ``planted partition model.'' Given a sparse stochastic block model, the two standard inference tasks are: (i) Weak recovery: can we estimate the communities with non trivial overlap with the true commu...