Networks are ubiquitous in science and have become a focal point for discussion in everyday life. Formal statistical models for the analysis of network data have emerged as a major topic of interest in diverse areas of study, and most of these involve a form of graphical representation. Probability models on graphs date back to 1959. Along with empirical studies in social psychology and sociology from the 1960s, these early works generated an active network community and a substantial literature in the 1970s. This effort moved into the statistical literature in the late 1970s and 1980s, and the past decade has seen a burgeoning network literature in statistical physics and computer science. The growth of the World Wide Web and the emergence of online networking communities such as Facebook, MySpace, and LinkedIn, and a host of more specialized professional network communities has intensified interest in the study of networks and network data. Our goal in this review is to provide the reader with an entry point to this burgeoning literature. We begin with an overview of the historical development of statistical network modeling and then we introduce a number of examples that have been studied in the network literature. Our subsequent discussion focuses on a number of prominent static and dynamic network models and their interconnections. We emphasize formal model descriptions, and pay special attention to the interpretation of parameters and their estimation. We end with a description of some open problems and challenges for machine learning and statistics.

In this paper, we propose causality as a unified framework to explain query answers and non-answers, thus generalizing and extending several previously proposed approaches of provenance and missing query result explanations. We develop our framework starting from the well-studied definition of actual causes by Halpern and Pearl [13]. After identifying some undesirable characteristics of the original definition, we propose functional causes as a refined definition of causality with several desirable properties. These properties allow us to apply our notion of causality in a database context and apply it uniformly to define the causes of query results and their individual contributions in several ways: (i) we can model both provenance as well as nonanswers, (ii) we can define explanations as either data in the input relations or relational operations in a query plan, and (iii) we can give graded degrees of responsibility to individual causes, thus allowing us to rank causes. In particular, our approach allows us to explain contributions to relational aggregate functions and to rank causes according to their respective responsibilities. We give complexity results and describe polynomial algorithms for evaluating causality in tractable cases. Throughout the paper, we illustrate the applicability of our framework with several examples. Overall, we develop in this paper the theoretical foundations of causality theory in a database context.

This paper presents several new tractability results for planning based on macros. We describe an algorithm that optimally solves planning problems in a class that we call inverted tree reducible, and is provably tractable for several subclasses of this class. By using macros to store partial plans that recur frequently in the solution, the algorithm is polynomial in time and space even for exponentially long plans. We generalize the inverted tree reducible class in several ways and describe modifications of the algorithm to deal with these new classes. Theoretical results are validated in experiments.

It is natural to ask: what kinds of matrices satisfy the Restricted Eigenvalue (RE) condition? In this paper, we associate the RE condition (Bickel-Ritov-Tsybakov 09) with the complexity of a subset of the sphere in $\R^p$, where $p$ is the dimensionality of the data, and show that a class of random matrices with independent rows, but not necessarily independent columns, satisfy the RE condition, when the sample size is above a certain lower bound. Here we explicitly introduce an additional covariance structure to the class of random matrices that we have known by now that satisfy the Restricted Isometry Property as defined in Candes and Tao 05 (and hence the RE condition), in order to compose a broader class of random matrices for which the RE condition holds. In this case, tools from geometric functional analysis in characterizing the intrinsic low-dimensional structures associated with the RE condition has been crucial in analyzing the sample complexity and understanding its statistical implications for high dimensional data.

There has been a lot of recent work on Bayesian methods for reinforcement learning exhibiting near-optimal online performance. The main obstacle facing such methods is that in most problems of interest, the optimal solution involves planning in an infinitely large tree. However, it is possible to obtain stochastic lower and upper bounds on the value of each tree node. This enables us to use stochastic branch and bound algorithms to search the tree efficiently. This paper proposes two such algorithms and examines their complexity in this setting.

The problem is sequence prediction in the following setting. A sequence $x_1,...,x_n,...$ of discrete-valued observations is generated according to some unknown probabilistic law (measure) $\mu$. After observing each outcome, it is required to give the conditional probabilities of the next observation. The measure $\mu$ belongs to an arbitrary but known class $C$ of stochastic process measures. We are interested in predictors $\rho$ whose conditional probabilities converge (in some sense) to the "true" $\mu$-conditional probabilities if any $\mu\in C$ is chosen to generate the sequence. The contribution of this work is in characterizing the families $C$ for which such predictors exist, and in providing a specific and simple form in which to look for a solution. We show that if any predictor works, then there exists a Bayesian predictor, whose prior is discrete, and which works too. We also find several sufficient and necessary conditions for the existence of a predictor, in terms of topological characterizations of the family $C$, as well as in terms of local behaviour of the measures in $C$, which in some cases lead to procedures for constructing such predictors. It should be emphasized that the framework is completely general: the stochastic processes considered are not required to be i.i.d., stationary, or to belong to any parametric or countable family.

Traditional staging is based on a formal approach of similarity leaning on dramaturgical ontologies and instanciation variations. Inspired by interactive data mining, that suggests different approaches, we give an overview of computer science and theater researches using computers as partners of the actor to escape the a priori specification of roles.

This paper extends k-means algorithms from the Euclidean domain to the domain of graphs. To recompute the centroids, we apply subgradient methods for solving the optimization-based formulation of the sample mean of graphs. To accelerate the k-means algorithm for graphs without trading computational time against solution quality, we avoid unnecessary graph distance calculations by exploiting the triangle inequality of the underlying distance metric following Elkan's k-means algorithm proposed in \cite{Elkan03}. In experiments we show that the accelerated k-means algorithm are faster than the standard k-means algorithm for graphs provided there is a cluster structure in the data.

This paper formulates a necessary and sufficient condition for a generic graph matching problem to be equivalent to the maximum vertex and edge weight clique problem in a derived association graph. The consequences of this results are threefold: first, the condition is general enough to cover a broad range of practical graph matching problems; second, a proof to establish equivalence between graph matching and clique search reduces to showing that a given graph matching problem satisfies the proposed condition; and third, the result sets the scene for generic continuous solutions for a broad range of graph matching problems. To illustrate the mathematical framework, we apply it to a number of graph matching problems, including the problem of determining the graph edit distance.

In the naming game, individuals or agents exchange pairwise local information in order to communicate about objects in their common environment. The goal of the game is to reach a consensus about naming these objects. Originally used to investigate language formation and self-organizing vocabularies, we extend the classical naming game with a globally shared memory accessible by all agents. This shared memory can be interpreted as an external source of knowledge like a book or an Internet site. The extended naming game models an environment similar to one that can be found in the context of social bookmarking and collaborative tagging sites where users tag sites using appropriate labels, but also mimics an important aspect in the field of human-based image labeling. Although the extended naming game is non-deterministic in its word selection, we show that consensus towards a common vocabulary is reached. More importantly, we show the qualitative and quantitative influence of the external source of information, i.e. the shared memory, on the consensus dynamics between the agents.