The stochastic blockmodel (SBM) models the connectivity within and between disjoint subsets of nodes in networks. Prior work demonstrated that the rows of an SBM's adjacency spectral embedding (ASE) and Laplacian spectral embedding (LSE) both converge in law to Gaussian mixtures where the components are curved exponential families. Maximum likelihood estimation via the Expectation-Maximization (EM) algorithm for a full Gaussian mixture model (GMM) can then perform the task of clustering graph nodes, albeit without appealing to the components' curvature. Noting that EM is a special case of the Expectation-Solution (ES) algorithm, we propose two ES algorithms that allow us to take full advantage of these curved structures. After presenting the ES algorithm for the general curved-Gaussian mixture, we develop those corresponding to the ASE and LSE limiting distributions. Simulating from artificial SBMs and a brain connectome SBM reveals that clustering graph nodes via our ES algorithms can improve upon that of EM for a full GMM for a wide range of settings.
We focus on spectral clustering of unlabeled graphs and review some results on clustering methods which achieve weak or strong consistent identification in data generated by such models. We also present a new algorithm which appears to perform optimally both theoretically using asymptotic theory and empirically.
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 choice of the tuning parameter. Moreover, our results show how the "star shape" in the eigenvectors--a common feature of…
Sri Sathya Sai Institute of Higher Learning（斯里·萨提亚·赛高等教育学院）
Spectral Clustering(SC) is a prominent data clustering technique of recent times which has attracted much attention from researchers. It is a highly data-driven method and makes no strict assumptions on the structure of the data to be clustered. One of the central pieces of spectral clustering is the construction of an affinity matrix based on a similarity measure between data points. The way the similarity measure is defined between data points has a direct impact on the performance of the SC technique. Several attempts have been made in the direction of strengthening the pairwise similarity …
Spectral clustering is a broad class of clustering procedures in which an intractable combinatorial optimization formulation of clustering is "relaxed" into a tractable eigenvector problem, and in which the relaxed solution is subsequently "rounded" into an approximate discrete solution to the original problem. In this paper we present a novel margin-based perspective on multiway spectral clustering. We show that the margin-based perspective illuminates both the relaxation and rounding aspects of spectral clustering, providing a unified analysis of existing algorithms and guiding the design of…
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 classify all individuals into the true community uniformly correctly almost surely. Extensions to both regularized spectral …
Local spectral clustering for heterogeneous clustering structures
School of Economics and Management, Fuzhou University（福州大学经济与管理学院）；School of Economics, Xiamen University（厦门大学经济学院）；The Wang Yanan Institute for Studies in Economics, Xiamen University（厦门大学王亚南经济研究院）；Yau Mathematical Sciences Center, Tsinghua University（清华大学丘成桐数学科学中心）；Beijing Institute of Mathematical Sciences and Applications（北京应用数学与计算物理研究所）
Mixed membership community detection is a challenge problem in network analysis. To estimate the memberships and study the impact of regularized spectral clustering under the mixed membership stochastic block (MMSB) model, this article proposes two efficient spectral clustering approaches based on regularized Laplacian matrix, Simplex Regularized Spectral Clustering (SRSC) and Cone Regularized Spectral Clustering (CRSC). SRSC and CRSC methods are designed based on the ideal simplex structure and the ideal cone structure in the variants of the eigen-decomposition of the population regularized L…
Center for Modern Statistics, School of Mathematics, Northwest University（西北大学数学学院现代统计中心）
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 developed sketching algorithms to obtain two randomized spectral clustering algorithms, namely, the random projection-ba…
In the present paper, we studied a Dynamic Stochastic Block Model (DSBM) under the assumptions that the connection probabilities, as functions of time, are smooth and that at most $s$ nodes can switch their class memberships between two consecutive time points. We estimate the edge probability tensor by a kernel-type procedure and extract the group memberships of the nodes by spectral clustering. The procedure is computationally viable, adaptive to the unknown smoothness of the functional connection probabilities, to the rate $s$ of membership switching and to the unknown number of clusters. I…
University of Wisconsin（威斯康星大学）；Johns Hopkins University（约翰斯·霍普金斯大学）
Biological and social systems consist of myriad interacting units. The interactions can be represented in the form of a graph or network. Measurements of these graphs can reveal the underlying structure of these interactions, which provides insight into the systems that generated the graphs. Moreover, in applications such as connectomics, social networks, and genomics, graph data are accompanied by contextualizing measures on each node. We utilize these node covariates to help uncover latent communities in a graph, using a modification of spectral clustering. Statistical guarantees are provide…
Directed networks are broadly used to represent asymmetric relationships among units. Co-clustering aims to cluster the senders and receivers of directed networks simultaneously. In particular, the well-known spectral clustering algorithm could be modified as the spectral co-clustering to co-cluster directed networks. However, large-scale networks pose great computational challenges to it. In this paper, we leverage sketching techniques and derive two randomized spectral co-clustering algorithms, one \emph{random-projection-based} and the other \emph{random-sampling-based}, to accelerate the c…
