How does the Web look? How could we tell an abnormal social network from a normal one? These and similar questions are important in many fields where the data can intuitively be cast as a graph; examples range from computer networks to sociology to biology and many more. Indeed, any M : N relation in database terminology can be represented as a graph. A lot of these questions boil down to the following: "How can we generate synthetic but realistic graphs?" To answer this, we must first understand what patterns are common in real-world graphs and can thus be considered a mark of normality/realism. This survey give an overview of the incredible variety of work that has been done on these problems. One of our main contributions is the integration of points of view from physics, mathematics, sociology, and computer science. Further, we briefly describe recent advances on some related and interesting graph problems.

We present a novel unsupervised learning scheme that simultaneously clusters variables of several types (e.g., documents, words and authors) based on pairwise interactions between the types, as observed in co-occurrence data. In this scheme, multiple clustering systems are generated aiming at maximizing an objective function that measures multiple pairwise mutual information between cluster variables. To implement this idea, we propose an algorithm that interleaves top-down clustering of some variables and bottom-up clustering of the other variables, with a local optimization correction routine. Focusing on document clustering we present an extensive empirical study of two-way, three-way and four-way applications of our scheme using six real-world datasets including the 20 Newsgroups (20NG) and the Enron email collection. Our multi-way distributional clustering (MDC) algorithms consistently and significantly outperform previous state-of-the-art information theoretic clustering algorithms. 1.

Abstract—We propose a novel nonnegative matrix factorization model that aims at finding localized, part-based, representations of nonnegative multivariate data items. Unlike the classical nonnegative matrix factorization (NMF) technique, this new model, denoted “nonsmooth nonnegative matrix factorization ” (nsNMF), corresponds to the optimization of an unambiguous cost function designed to explicitly represent sparseness, in the form of nonsmoothness, which is controlled by a single parameter. In general, this method produces a set of basis and encoding vectors that are not only capable of representing the original data, but they also extract highly localized patterns, which generally lend themselves to improved interpretability. The properties of this new method are illustrated with several data sets. Comparisons to previously published methods show that the new nsNMF method has some advantages in keeping faithfulness to the data in the achieving a high degree of sparseness for both the estimated basis and the encoding vectors and in better interpretability of the factors. Index Terms—nonnegative matrix factorization, constrained optimization, datamining, mining methods and algorithms, pattern analysis, feature extraction or construction, sparse, structured, and very large systems. æ 1

While the vast majority of clustering algorithms are partitional, many real world datasets have inherently overlapping clusters. Several approaches to finding overlapping clusters have come from work on analysis of biological datasets. In this paper, we interpret an overlapping clustering model proposed by Segal et al. [23] as a generalization of Gaussian mixture models, and we extend it to an overlapping clustering model based on mixtures of any regular exponential family distribution and the corresponding Bregman divergence. We provide the necessary algorithm modifications for this extension, and present results on synthetic data as well as subsets of 20-Newsgroups and EachMovie datasets.

In this paper we introduce a novel algorithm called triCluster, for mining coherent clusters in three-dimensional (3D) gene expression datasets. triCluster can mine arbitrarily positioned and overlapping clusters, and depending on di#erent parameter values, it can mine di#erent types of clusters, including those with constant or similar values along each dimension, as well as scaling and shifting expression patterns. triCluster relies on graph-based approach to mine all valid clusters. For each time slice, i.e., a genesample matrix, it constructs the range multigraph, a compact representation of all similar value ranges between any two sample columns. It then searches for constrained maximal cliques in this multigraph to yield the set of biclusters for this time slice. Then triCluster constructs another graph using the biclusters (as vertices) from each time slice; mining cliques from this graph yields the final set of triclusters. Optionally, triCluster merges/deletes some clusters having large overlaps. We present a useful set of metrics to evaluate the clustering quality, and we show that triCluster can find significant triclusters in the real microarray datasets.

A number of real-world domains such as social networks and e-commerce involve heterogeneous data that describes relations between multiple classes of entities. Understanding the natural structure of this type of heterogeneous relational data is essential both for exploratory analysis and for performing various predictive modeling tasks. In this paper, we propose a principled multi-way clustering framework for relational data, wherein different types of entities are simultaneously clustered based not only on their intrinsic attribute values, but also on the multiple relations between the entities. To achieve this, we introduce a relation graph model that describes all the known relations between the different entity classes, in which each relation between a given set of entity classes is represented in the form of multi-modal tensor over an appropriate domain. Our multi-way clustering formulation is driven by the objective of capturing the maximal “information ” in the original relation graph, i.e., accurately approximating the set of tensors corresponding to the various relations. This formulation is applicable to all Bregman divergences (a broad family of loss functions that includes squared Euclidean distance, KL-divergence), and also permits analysis of mixed data types using convex combinations of appropriate Bregman loss functions. Furthermore, we present a large family of structurally different multi-way clustering schemes that preserve various linear summary statistics of the original data. We accomplish the above generalizations by extending a recently proposed key theoretical result, namely the minimum Bregman information principle [1], to the relation graph setting. We also describe an efficient multi-way clustering algorithm based on alternate minimization that generalizes a number of other recently proposed clustering methods. Empirical results on datasets obtained from real-world domains (e.g., movie recommendations, newsgroup articles) demonstrate the generality and efficacy of our framework. 1

Huge datasets are becoming prevalent; even as researchers, we now routinely have to work with datasets that are up to a few terabytes in size. Interesting real-world applications produce huge volumes of messy data. The mining process involves several steps, starting from pre-processing the raw data to estimating the final models. As data become more abundant, scalable and easyto-use tools for distributed processing are also emerging. Among those, Map-Reduce has been widely embraced by both academia and industry. In database terms, Map-Reduce is a simple yet powerful execution engine, which can be complemented with other data storage and management components, as necessary. In this paper we describe our experiences and findings in applying Map-Reduce, from raw data to final models, on an important mining task. In particular, we focus on co-clustering, which has been studied in many applications such as text mining, collaborative filtering, bio-informatics, graph mining. We propose the Distributed Co-clustering (DisCo) framework, which introduces practical approaches for distributed data pre-processing, and co-clustering. We develop DisCo using Hadoop, an open source Map-Reduce implementation. We show that DisCo can scale well and efficiently process and analyze extremely large datasets (up to several hundreds of gigabytes) on commodity hardware. 1

We consider the Discrete Basis Problem, which can be described as follows: given a collection of Boolean vectors find a collection of k Boolean basis vectors such that the original vectors can be represented using disjunctions of these basis vectors. We show that the decision version of this problem is NP-complete and that the optimization version cannot be approximated within any finite ratio. We also study two variations of this problem, where the Boolean basis vectors must be mutually otrhogonal. We show that the other variation is closely related with the well-known Metric k-median Problem in Boolean space. To solve these problems, two algorithms will be presented. One is designed for the variations mentioned above, and it is solely based on solving the k-median problem, while another is a heuristic intended to solve the general Discrete Basis Problem. We will also study the results of extensive experiments made with these two algorithms with both synthetic and real-world data. The results are twofold: with the synthetic data, the algorithms did rather well, but with the real-world data the results were not as good.

We present a general model-independent approach to the analysis of data in cases when these data do not appear in the form of co-occurrence of two variables X, Y, but rather as a sample of values of an unknown (stochastic) function Z(X, Y). For example, in gene expression data, the expression level Z is a function of gene X and condition Y; or in movie ratings data the rating Z is a function of viewer X and movie Y. The approach represents a consistent extension of the Information Bottleneck method that has previously relied on the availability of co-occurrence statistics. By altering the relevance variable we eliminate the need in the sample of joint distribution of all input variables. This new formulation also enables simple MDL-like model complexity control and prediction of missing values of Z. The approach is analyzed and shown to be on a par with the best known clustering algorithms for a wide range of domains. For the prediction of missing values (collaborative filtering) it improves the currently best known results. 1

We propose a new formulation of the conceptual clustering problem where the goal is to explicitly output a collection of simple and meaningful conjunctions of attributes that define the clusters. The formulation differs from previous approaches since the clusters discovered may overlap and also may not cover all the points. In addition, a point may be assigned to a cluster description even if it only satisfies most, and not necessarily all, of the attributes in the conjunction. Connections between this conceptual clustering problem and the maximum edge biclique problem are made. Simple, randomized algorithms are given that discover a collection of approximate conjunctive cluster descriptions in sublinear time.