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## Statistical Analysis of Graph Structures in Random Variable Networks

This book studies complex systems with elements represented by random variables. Its main goal is to study and compare uncertainty of algorithms of network structure identification with applications to market network analysis. For this, a mathematical model of random variable network is introduced, uncertainty of identification procedure is defined through a risk function, random variables networks with different measures of similarity (dependence) are discussed, and general statistical properties of identification algorithms are studied. The volume also introduces a new class of identification algorithms based on a new measure of similarity and prove its robustness in a large class of distributions, and presents applications to social networks, power transmission grids, telecommunication networks, stock market networks, and brain networks through a theoretical analysis that identifies network structures. Both researchers and graduate students in computer science, mathematics, and optimization will find the applications and techniques presented useful.

The paper deal with uncertainty in market network analysis. The main problem addressed is to investigate statistical uncertainty of Kruskal algorithm for the minimum spanning tree in market network. Uncertainty of Kruskal algorithm is measured by the probability of q incorrectly included edges. Numerical experiments are conducted with the returns of a set of 100 financial instruments traded in the US stock market over a period of 250 days in 2014. Obtained results help to estimate the reliability of minimum spanning tree in market network analysis.

This book highlights cutting-edge research in the field of network science, offering scientists, researchers, students and practitioners a unique update on the latest advances in theory, together with a wealth of applications. It presents the peer-reviewed proceedings of the VII International Conference on Complex Networks and their Applications (COMPLEX NETWORKS 2018), which was held in Cambridge on December 11–13, 2018. The carefully selected papers cover a wide range of theoretical topics such as network models and measures; community structure and network dynamics; diffusion, epidemics and spreading processes; and resilience and control; as well as all the main network applications, including social and political networks; networks in finance and economics; biological and neuroscience networks; and technological networks.

Problem of construction of the market graph as a multiple decision statistical problem is considered. Detailed description of a optimal unbiased multiple decision statistical procedure is given. This procedure is constructed using the Lehmann’s theory of multiple decision statistical procedures and the conditional tests of the Neyman structures. The equations for thresholds calculation for the tests of the Neyman structure are presented and analyzed.

This volume contains two types of papers—a selection of contributions from the “Second International Conference in Network Analysis” held in Nizhny Novgorod on May 7–9, 2012, and papers submitted to an "open call for papers" reflecting the activities of LATNA at the Higher School for Economics.

This volume contains many new results in modeling and powerful algorithmic solutions applied to problems in

- vehicle routing

- single machine scheduling

- modern financial markets

- cell formation in group technology

- brain activities of left- and right-handers

- speeding up algorithms for the maximum clique problem

- analysis and applications of different measures in clustering

The broad range of applications that can be described and analyzed by means of a network brings together researchers, practitioners, and other scientific communities from numerous fields such as Operations Research, Computer Science, Bioinformatics, Medicine, Transportation, Energy, Social Sciences, and more. The contributions not only come from different fields, but also cover a broad range of topics relevant to the theory and practice of network analysis. Researchers, students, and engineers from various disciplines will benefit from the state-of-the-art in models, algorithms, technologies, and techniques including new research directions and open questions.

Research into the market graph is attracting increasing attention in stock market analysis. One of the important problems connected with the market graph is its identification from observations. The standard way of identifying the market graph is to use a simple procedure based on statistical estimations of Pearson correlations between pairs of stocks. Recently a new class of statistical procedures for market graph identification was introduced and the optimality of these procedures in the Pearson correlation Gaussian network was proved. However, the procedures obtained have a high reliability only for Gaussian multivariate distributions of stock attributes. One of the ways to correct this problem is to consider different networks generated by different measures of pairwise similarity of stocks. A new and promising model in this context is the sign similarity network. In this paper the market graph identification problem in the sign similarity network is reviewed. A new class of statistical procedures for the market graph identification is introduced and the optimality of these procedures is proved. Numerical experiments reveal an essential difference in the quality between optimal procedures in sign similarity and Pearson correlation networks. In particular, it is observed that the quality of the optimal identification procedure in the sign similarity network is not sensitive to the assumptions on the distribution of stock attributes.

Problem of multiple comparisons of several populations on small samples and specificity of the method of it solution are analyzed. It is proposed to extend a classical method for constructing statistical tests by the use of information preprocessing. Examples of the application of the proposed method are given.

Market network analysis attracts a growing attention last decade. Important component of the market network is a model of stock returns distribution. Elliptically contoured distributions are popular as probability model of stock returns. The question of adequacy of this model to real market data is open. There are known results that reject such model and at the same time there are results that approve such model. Obtained results are concerned to testing some properties of elliptical model. In the paper another property of elliptical model namely property of symmetry condition of tails of 2-dimentional distribution is considered. Multiple statistical procedure for testing elliptical model for stock returns distribution is proposed. Sign symmetry conditions of tails distribution are chosen as individual hypotheses for multiple testing. Uniformly most powerful tests of Neyman structure are constructed for individual hypotheses testing. Associated stepwise multiple testing procedure is applied for the real market data. To visualize the results a rejection graph is constructed. The main result is that under some conditions tail symmetry hypothesis is not rejected if one remove a few number of hubs from the rejection graph.

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.