Spectral Properties and Weak Detection in Stochastic Block Models

Community detection in networks is a key exploratory tool with applications in a diverse set of areas, ranging from finding communities in social and biological networks to identifying link farms in the World Wide Web. The problem of finding communities or clusters in a network has received much attention from statistics, physics and computer science. However, most clustering algorithms assume knowledge of the number of clusters k. In this paper we propose to automatically de...

Detecting communities in high-dimensional graphs can be achieved by applying random matrix theory where the adjacency matrix of the graph is modeled by a Stochastic Block Model (SBM). However, the SBM makes an unrealistic assumption that the edge probabilities are homogeneous within communities, i.e., the edges occur with the same probabilities. The Degree-Corrected SBM is a generalization of the SBM that allows these edge probabilities to be different, but existing results f...

We study the problem of detecting the community structure from the generalized stochastic block model (GSBM). Based on the analysis of the Stieljtes transform of the empirical spectral distribution, we prove a BBP-type transition for the largest eigenvalue of the GSBM. For specific models such as a hidden community model and an unbalanced stochastic model, we provide precise formulas for the two largest eigenvalues, establishing the gap in the BBP-type transition.

Community detection in complex networks has attracted considerable attention, however, most existing methods need the number of communities to be specified beforehand. In this paper, a goodness-of-fit test based on the linear spectral statistic of the centered and rescaled adjacency matrix for the stochastic block model is proposed. We prove that the proposed test statistic converges in distribution to the standard Gaussian distribution under the null hypothesis. The proof us...

In this paper we prove the strong consistency of several methods based on the spectral clustering techniques that are widely used to study the community detection problem in stochastic block models (SBMs). We show that under some weak conditions on the minimal degree, the number of communities, and the eigenvalues of the probability block matrix, the K-means algorithm applied to the eigenvectors of the graph Laplacian associated with its first few largest eigenvalues can clas...

The stochastic block model (SBM) is a random graph model with different group of vertices connecting differently. It is widely employed as a canonical model to study clustering and community detection, and provides a fertile ground to study the information-theoretic and computational tradeoffs that arise in combinatorial statistics and more generally data science. This monograph surveys the recent developments that establish the fundamental limits for community detection in...

Spectral clustering has been widely used for community detection in network sciences. While its empirical successes are well-documented, a clear theoretical understanding, particularly for sparse networks where degrees are much smaller than $\log n$, remains unclear. In this paper, we address this significant gap by demonstrating that spectral clustering offers exponentially small error rates when applied to sparse networks under Stochastic Block Models. Our analysis provides...

The present work is concerned with community detection. Specifically, we consider a random graph drawn according to the stochastic block model~: its vertex set is partitioned into blocks, or communities, and edges are placed randomly and independently of each other with probability depending only on the communities of their two endpoints. In this context, our aim is to recover the community labels better than by random guess, based only on the observation of the graph. In t...

Motivated by community detection, we characterise the spectrum of the non-backtracking matrix $B$ in the Degree-Corrected Stochastic Block Model. Specifically, we consider a random graph on $n$ vertices partitioned into two equal-sized clusters. The vertices have i.i.d. weights $\{ \phi_u \}_{u=1}^n$ with second moment $\Phi^{(2)}$. The intra-cluster connection probability for vertices $u$ and $v$ is $\frac{\phi_u \phi_v}{n}a$ and the inter-cluster connection probability is...

We consider community detection in Degree-Corrected Stochastic Block Models (DC-SBM). We propose a spectral clustering algorithm based on a suitably normalized adjacency matrix. We show that this algorithm consistently recovers the block-membership of all but a vanishing fraction of nodes, in the regime where the lowest degree is of order log$(n)$ or higher. Recovery succeeds even for very heterogeneous degree-distributions. The used algorithm does not rely on parameters as i...