Hypothesis Testing For Community Detection On The Stochastic Block Model

This self-contained, compact monograph is an invaluable introduction to the field of Community Detection for researchers and students working in Machine Learning, Data Science and Information Theory.

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 the SBM, both with respect to information-theoretic and computational tradeoffs, and for various recovery requirements such as exact, partial and weak recovery. The main results discussed are the phase transitions for exact recovery at the Chernoff-Hellinger threshold, the phase transition for weak recovery at the Kesten-Stigum threshold, the optimal SNR-mutual information tradeoff for partial recovery, and the gap between information-theoretic and computational thresholds.

Provides an overview of the developments and advances in the field of network clustering and blockmodeling over the last 10 years This book offers an integrated treatment of network clustering and blockmodeling, covering all of the newest approaches and methods that have been developed over the last decade. Presented in a comprehensive manner, it offers the foundations for understanding network structures and processes, and features a wide variety of new techniques addressing issues that occur during the partitioning of networks across multiple disciplines such as community detection, blockmodeling of valued networks, role assignment, and stochastic blockmodeling. Written by a team of international experts in the field, Advances in Network Clustering and Blockmodeling offers a plethora of diverse perspectives covering topics such as: bibliometric analyses of the network clustering literature; clustering approaches to networks; label propagation for clustering; and treating missing network data before partitioning. It also examines the partitioning of signed networks, multimode networks, and linked networks. A chapter on structured networks and coarsegrained descriptions is presented, along with another on scientific coauthorship networks. The book finishes with a section covering conclusions and directions for future work. In addition, the editors provide numerous tables, figures, case studies, examples, datasets, and more. Offers a clear and insightful look at the state of the art in network clustering and blockmodeling Provides an excellent mix of mathematical rigor and practical application in a comprehensive manner Presents a suite of new methods, procedures, algorithms for partitioning networks, as well as new techniques for visualizing matrix arrays Features numerous examples throughout, enabling readers to gain a better understanding of research methods and to conduct their own research effectively Written by leading contributors in the field of spatial networks analysis Advances in Network Clustering and Blockmodeling is an ideal book for graduate and undergraduate students taking courses on network analysis or working with networks using real data. It will also benefit researchers and practitioners interested in network analysis.

The aim of the book is to introduce basic concepts, main results, and widely applied mathematical tools in the spectral analysis of large dimensional random matrices. The core of the book focuses on results established under moment conditions on random variables using probabilistic methods, and is thus easily applicable to statistics and other areas of science. The book introduces fundamental results, most of them investigated by the authors, such as the semicircular law of Wigner matrices, the Marcenko-Pastur law, the limiting spectral distribution of the multivariate F matrix, limits of extreme eigenvalues, spectrum separation theorems, convergence rates of empirical distributions, central limit theorems of linear spectral statistics, and the partial solution of the famous circular law. While deriving the main results, the book simultaneously emphasizes the ideas and methodologies of the fundamental mathematical tools, among them being: truncation techniques, matrix identities, moment convergence theorems, and the Stieltjes transform. Its treatment is especially fitting to the needs of mathematics and statistics graduate students and beginning researchers, having a basic knowledge of matrix theory and an understanding of probability theory at the graduate level, who desire to learn the concepts and tools in solving problems in this area. It can also serve as a detailed handbook on results of large dimensional random matrices for practical users. This second edition includes two additional chapters, one on the authors' results on the limiting behavior of eigenvectors of sample covariance matrices, another on applications to wireless communications and finance. While attempting to bring this edition up-to-date on recent work, it also provides summaries of other areas which are typically considered part of the general field of random matrix theory.

This volume includes the main contributions by the plenary speakers from the ISAAC congress held in Aveiro, Portugal, in 2019. It is the purpose of ISAAC to promote analysis, its applications, and its interaction with computation. Analysis is understood here in the broad sense of the word, including differential equations, integral equations, functional analysis, and function theory. With this objective, ISAAC organizes international Congresses for the presentation and discussion of research on analysis. The plenary lectures in the present volume, authored by eminent specialists, are devoted to some exciting recent developments in topics such as science data, interpolating and sampling theory, inverse problems, and harmonic analysis.

This book provides an integrated treatment of generalized blockmodeling appropriate for the analysis network structures.

In many sciences---for example Sociology, Biology, and Computer Science---units under study often belong to communities, and units within the same community behave similarly. A way to identify communities may be through how units interact with each other; units within a community may be more likely to interact with each other than units across different communities. This is equivalent to viewing units under study as a network, where nodes are units and edges are drawn between two units if they interact with each other. Hence, a critical problem in these sciences is how to identify communities given a mathematical network. Since members within the same community are more likely to interact with each other, it may follow that cycles may be more prevalent within communities than across communities. Thus, the detection of these communities may be aided through the use of measures of the local ``richness'' of cyclic structures. In this dissertation, we develop the renewal non-backtracking random walk (RNBRW)--- a variant of a random walk in which the walk is prohibited from returning back to a node in exactly two steps and terminates and restarts once it completes a loop---as a way of quantifying this cyclic structure. Specifically, we propose using the retracing probability of an edge---the likelihood that the edge completes a cycle in a RNBRW---as a way of quantifying cyclic structure. Intuitively, edges with larger retracing probabilities should be more important to the formation of cycles, and hence, to the detection of communities. We show that retracing probabilities can be estimated efficiently through repeated iterations of RNBRW. Additionally, since RNBRW runs can be performed in parallel, accurate estimation can be obtained even when the network contains millions of nodes. We give simulation results that suggest pre-weighting edges through RNBRW can improve the performance of popular community detection algorithms substantially. This improvement is most significant when the network is sparse. Moreover, the performance of community detection algorithms with this weighting are competitive to other scalable methods that do not allow for weighting of edges. We also develop a goodness-of-fit test to help determine whether communities exist within a network. We begin with a network on $n$ nodes that follows an unknown random graph model. We test the null hypothesis that the network is a realization of an Erdös-Renyi graph---a random graph in which each edge is equally likely to be formed, and hence, contains no inherent community structure. Rejecting this null implies that the network comes from a distribution with inherent community structure (for example, a planted partition model). To perform our test, we form an $n\times n$ matrix where entries are the retracing probabilities (estimated through RNBRW) of the corresponding edges of the network. We use as our test statistic a scaled version of the largest eigenvalue of this matrix. We perform a simulation study to compare the Type I Error probability and power of our method to that of other spectral approaches for network inference. We conclude by describing connections between RNBRW and the maximum expected cyclic overlap problem and giving theoretical results of RNBRW under the stochastic block model.

Graph-structured data is ubiquitous throughout the natural and social sciences, from telecommunication networks to quantum chemistry. Building relational inductive biases into deep learning architectures is crucial for creating systems that can learn, reason, and generalize from this kind of data. Recent years have seen a surge in research on graph representation learning, including techniques for deep graph embeddings, generalizations of convolutional neural networks to graph-structured data, and neural message-passing approaches inspired by belief propagation. These advances in graph representation learning have led to new state-of-the-art results in numerous domains, including chemical synthesis, 3D vision, recommender systems, question answering, and social network analysis. This book provides a synthesis and overview of graph representation learning. It begins with a discussion of the goals of graph representation learning as well as key methodological foundations in graph theory and network analysis. Following this, the book introduces and reviews methods for learning node embeddings, including random-walk-based methods and applications to knowledge graphs. It then provides a technical synthesis and introduction to the highly successful graph neural network (GNN) formalism, which has become a dominant and fast-growing paradigm for deep learning with graph data. The book concludes with a synthesis of recent advancements in deep generative models for graphs—a nascent but quickly growing subset of graph representation learning.