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## Network partitioning algorithms as cooperative games

The paper is devoted to game-theoretic methods for community detection in networks. The traditional methods for detecting community structure are based on selecting dense subgraphs inside the network. Here we propose to use the methods of cooperative game theory that highlight not only the link density but also the mechanisms of cluster formation. Specifically, we suggest two approaches from cooperative game theory: the first approach is based on the Myerson value, whereas the second approach is based on hedonic games. Both approaches allow to detect clusters with various resolutions. However, the tuning of the resolution parameter in the hedonic games approach is particularly intuitive. Furthermore, the modularity-based approach and its generalizations as well as ratio cut and normalized cut methods can be viewed as particular cases of the hedonic games. Finally, for approaches based on potential hedonic games we suggest a very efficient computational scheme using Gibbs sampling.

Multimodal clustering is an unsupervised technique for mining interesting patterns in n-ary relations or n-mode networks. Among different types of such generalised patterns one can find biclusters and formal concepts (maximal bicliques) for two-mode case, triclusters and triconcepts for three-mode case, closed n-sets for n-mode case, etc. Object-attribute biclustering (OA-biclustering) for mining large binary datatables (formal contexts or two-mode networks) arose by the end of the last decade due to intractability of computation problems related to formal concepts; this type of patterns was proposed as a meaningful and scalable approximation of formal concepts. In this paper, our aim is to present recent advance in OA-biclustering and its extensions to mining multi-mode communities in SNA setting. We also discuss connection between clustering coefficients known in SNA community for one-mode and two-mode networks and OA-bicluster density, the main quality measure of an OA-bicluster. Our experiments with two-, three-, and four-mode large real-world networks show that this type of patterns is suitable for community detection in multi-mode cases within reasonable time even though the number of corresponding n-cliques is still unknown due to computation difficulties. An interpretation of OA-biclusters for one-mode networks is provided as well.

In this paper we introduce a generalized learning algorithm for probabilistic topic models (PTM). Many known and new algorithms for PLSA, LDA, and SWB models can be obtained as its special cases by choosing a subset of the following “options”: regularization, sampling, update frequency, sparsing and robustness. We show that a robust topic model, which distinguishes specific, background and topic terms, doesn’t need Dirichlet regularization and provides controllably sparse solution.

This paper examines two Markov chain Monte Carlo methods that have been widely used in econometrics. An introductory exposition of the Metropolis algorithm and the Gibbs sampler is provided. These methods are used to simulate multivariate distributions. Many problems in Bayesian statistics can be solved by simulating the posterior distribution. Invariance condition is of importance, the proofs are given for both methods. We use finite Markov chains to explore and substantiate the methods. Several examples are provided to illustrate the applicability and efficiency of the Markov chain Monte Carlo methods. They include bivariate normal distribution with high correlation, bivariate exponential distribution, mixture of bivariate normals.

**Importance** The paper is devoted to analysis of the effectiveness of economic integration of firms. By efficiency I mean the standard requirements for profitability of integration (non-decreasing of total profit) in microeconomics, theory of the firm and the theory of industrial organization, or non-negativity of synergy in the theory of corporate finance and business valuation theory.

**Objectives** The purpose of the article is to derive the fair value of companies within the economic integration. Its definition is necessary to take into account the effect of external interaction in the competitive environment on the value of the business, to determine the volume of shares exchanged in the merger, the synergy share to pay to the acquired company in the purchase price, for deciding about splitting the business.

**Methods **Each company aims to increase its own value, which indicates the conflicted nature of the interaction between agents. Thus, this paper proposes the use of tools of the cooperative game theory to determine the profitability of integration for each of the participating firms, taking into account non-decreasing its share in the fair value of the entire integration.

**Results **The paper formalizes the notion of profitability of economic integration and the fair value of companies with the tools of cooperative game theory. It proofs the interpretation of solution concept of cooperative game as a method of calculating the fair value of companies with regard to its external cooperative interaction with contractors or the purchase price of acquired companies in M&A deals. The paper provides an example of such an analysis for economic integration in the aviation industry.

**Conclusions and Relevance **The proposed approach to the analysis of economic integration extends the understanding of its nature, making it possible to estimate the contribution of each individual firm. The article is of practical importance for companies to jointly carry out R&D, supply chains, alliances, holdings, M&A deals or investment and consulting companies serving such transactions.

Interval cooperative games are models of cooperative situation where only bounds for payoffs of coalitions are known with certainty. The extension of solutions of classical cooperative games to interval setting highly depends on their monotonicity properties. However. both the prenucleolus and the tau-value are not aggregate monotonic on the class of convex TU games Hokari (2000, 2001). Therefore, interval analogues of these solutions either should be defined by another manner, or perhaps they exist in some other class of interval games. Both approaches are used in the paper: the prenucleolus of a convex interval game is defined by lexicographical minimization of the lexmin relation on the set of joint excess vectors of lower and upper games. On the other hand, the tau-value is shown to satisfy extendability condition on a subclass of convex games -- on the class of totally positive convex games. The interval prenucleolus is determined , and the proof of non-emptiness of the interval \tau-value on the class of interval totally positive games is given.

In this paper we introduce a generalized learning algorithm for probabilistic topic models (PTM). Many known and new algorithms for PLSA, LDA, and SWB models can be obtained as its special cases by choosing a subset of the following “options”: regularization, sampling, update frequency, sparsing and robustness. We show that a robust topic model, which distinguishes specific, background and topic terms, doesn’t need Dirichlet regularization and provides controllably sparse solution.

We consider certain spaces of functions on the circle, which naturally appear in harmonic analysis, and superposition operators on these spaces. We study the following question: which functions have the property that each their superposition with a homeomorphism of the circle belongs to a given space? We also study the multidimensional case.

We consider the spaces of functions on the m-dimensional torus, whose Fourier transform is p -summable. We obtain estimates for the norms of the exponential functions deformed by a C1 -smooth phase. The results generalize to the multidimensional case the one-dimensional results obtained by the author earlier in “Quantitative estimates in the Beurling—Helson theorem”, Sbornik: Mathematics, 201:12 (2010), 1811 – 1836.

We consider the spaces of function on the circle whose Fourier transform is p-summable. We obtain estimates for the norms of exponential functions deformed by a C1 -smooth phase.

This proceedings publication is a compilation of selected contributions from the “Third International Conference on the Dynamics of Information Systems” which took place at the University of Florida, Gainesville, February 16–18, 2011. The purpose of this conference was to bring together scientists and engineers from industry, government, and academia in order to exchange new discoveries and results in a broad range of topics relevant to the theory and practice of dynamics of information systems. Dynamics of Information Systems: Mathematical Foundation presents state-of-the art research and is intended for graduate students and researchers interested in some of the most recent discoveries in information theory and dynamical systems. Scientists in other disciplines may also benefit from the applications of new developments to their own area of study.