In standard clustering problems, data points are represented by vectors, and by stacking them together, one forms a data matrix with row or column cluster structure. In this paper, we consider a class of binary matrices, arising in many applications, which exhibit both row and column cluster structure, and our goal is to exactly recover the underlying row and column clusters by observing only a small fraction of noisy entries. We first derive a lower bound on the minimum number of observations needed for exact cluster recovery. Then, we propose three algorithms with different running time and compare the number of observations needed by them for successful cluster recovery. Our analytical results show smooth time-data trade-offs: one can gradually reduce the computational complexity when increasingly more observations are available.
Much of the prior work on graph clustering, as surveyed in @cite_30 , focuses on graphs with a single node type, where nodes in the same cluster are more likely to have edges among them. A low-rank plus sparse matrix decomposition approach is proved to exactly recover the clusters with the best known performance guarantee in @cite_13 . The same approach is used to recover the clusters from a partially observed graph in @cite_20 . A spectral method for exact cluster recovery is proposed and analyzed in @cite_6 with the number of clusters fixed . More recently, @cite_25 proved an upper bound on the number of nodes mis-clustered'' by a spectral clustering algorithm in the high dimensional setting with a growing number of clusters.
@cite_30 @cite_6 @cite_13 @cite_25 @cite_20
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The modern science of networks has brought significant advances to our understanding of complex systems. One of the most relevant features of graphs representing real systems is community structure, or clustering, i.e. the organization of vertices in clusters, with many edges joining vertices of the same cluster and comparatively few edges joining vertices of different clusters. Such clusters, or communities, can be considered as fairly independent compartments of a graph, playing a similar role like, e.g., the tissues or the organs in the human body. Detecting communities is of great importance in sociology, biology and computer science, disciplines where systems are often represented as graphs. This problem is very hard and not yet satisfactorily solved, despite the huge effort of a large interdisciplinary community of scientists working on it over the past few years. We will attempt a thorough exposition of the topic, from the definition of the main elements of the problem, to the presentation of most methods developed, with a special focus on techniques designed by statistical physicists, from the discussion of crucial issues like the significance of clustering and how methods should be tested and compared against each other, to the description of applications to real networks. Problems such as bisection, graph coloring, and clique are generally believed hard in the worst case. However, they can be solved if the input data is drawn randomly from a distribution over graphs containing acceptable solutions. In this paper we show that a simple spectral algorithm can solve all three problems above in the average case, as well as a more general problem of partitioning graphs based on edge density. In nearly all cases our approach meets or exceeds previous parameters, while introducing substantial generality. We apply spectral techniques, using foremost the observation that in all of these problems, the expected adjacency matrix is a low rank matrix wherein the structure of the solution is evident. We develop a new algorithm to cluster sparse unweighted graphs - i.e. partition the nodes into disjoint clusters so that there is higher density within clusters, and low across clusters. By sparsity we mean the setting where both the in-cluster and across cluster edge densities are very small, possibly vanishing in the size of the graph. Sparsity makes the problem noisier, and hence more difficult to solve. Any clustering involves a tradeoff between minimizing two kinds of errors: missing edges within clusters and present edges across clusters. Our insight is that in the sparse case, these must be penalized differently. We analyze our algorithm's performance on the natural, classical and widely studied "planted partition" model (also called the stochastic block model); we show that our algorithm can cluster sparser graphs, and with smaller clusters, than all previous methods. This is seen empirically as well. Networks or graphs can easily represent a diverse set of data sources that are characterized by interacting units or actors. Social ne tworks, representing people who communicate with each other, are one example. Communities or clusters of highly connected actors form an essential feature in the structure of several empirical networks. Spectral clustering is a popular and computationally feasi ble method to discover these communities. The Stochastic Block Model (, 1983) is a social network model with well defined communities; each node is a member of one community. For a network generated from the Stochastic Block Model, we bound the number of nodes "misclus- tered" by spectral clustering. The asymptotic results in th is paper are the first clustering results that allow the number of clusters in the model to grow with the number of nodes, hence the name high-dimensional. In order to study spectral clustering under the Stochastic Block Model, we first show that under the more general latent space model, the eigenvectors of the normalized graph Laplacian asymptotically converge to the eigenvectors of a "population" normal- ized graph Laplacian. Aside from the implication for spectral clustering, this provides insight into a graph visualization technique. Our method of studying the eigenvectors of random matrices is original. AMS 2000 subject classifications: Primary 62H30, 62H25; secondary 60B20. This paper considers the problem of clustering a partially observed unweighted graph---i.e., one where for some node pairs we know there is an edge between them, for some others we know there is no edge, and for the remaining we do not know whether or not there is an edge. We want to organize the nodes into disjoint clusters so that there is relatively dense (observed) connectivity within clusters, and sparse across clusters. We take a novel yet natural approach to this problem, by focusing on finding the clustering that minimizes the number of "disagreements"---i.e., the sum of the number of (observed) missing edges within clusters, and (observed) present edges across clusters. Our algorithm uses convex optimization; its basis is a reduction of disagreement minimization to the problem of recovering an (unknown) low-rank matrix and an (unknown) sparse matrix from their partially observed sum. We evaluate the performance of our algorithm on the classical Planted Partition Stochastic Block Model. Our main theorem provides sufficient conditions for the success of our algorithm as a function of the minimum cluster size, edge density and observation probability; in particular, the results characterize the tradeoff between the observation probability and the edge density gap. When there are a constant number of clusters of equal size, our results are optimal up to logarithmic factors.