### Article

## Robustness comparison of two market network models

Two market network models are investigated. One of them is based on the classical Pearson correlation

as the measure of association between stocks returns, whereas the second one is based on the

sign similarity measure of association between stocks returns. We study the uncertainty of

identification procedures for the following market network characteristics: distribution of weights

of edges, vertex degree distribution in the market graph, cliques and independent sets in the market

graph, and the vertex degree distribution of the maximum spanning tree. We define the true

network characteristics, the losses from the error of its identification by observations, and the uncertainty

of identification procedures as the expected value of losses. We use elliptically contoured distribution

as a model of multivariate stocks returns distribution. It is shown that identification statistical

procedures based on the sign similarity are statistically robust in contrast to the procedures based on the classical Pearson correlation

The main goal of the present paper is the development of general approach to network analysis of statistical data sets. First a general method of market network construction is proposed on the base of idea of measures of association. It is noted that many existing network models can be obtained as a particular case of this method. Next it is shown that statistical multiple decision theory is an appropriate theoretical basis for market network analysis of statistical data sets. Finally conditional risk for multiple decision statistical procedures is introduced as a natural measure of quality in market network analysis. Some illustrative examples are given.

Network model of stock market based on correlation matrix is considered. In the model vector of stock returns is supposed to hve multivariate normal distribution with given correlation matrix. Statistical uncertainty of some popular market network structures is analyzed by numerical simulation for network models of stock makets for different countries. For each market statistical uncertainty of different structures is compared. It is observed that despite of diversity the results of comparison are nearly the same for different markets. This leads to conjecture that there is some unknown common feature in different market network.

A class of distribution free multiple decision statistical procedures is proposed for threshold graph identification in correlation networks. The decision procedures are based on simultaneous application of sign statistics. It is proved that single step, step down Holm and step up Hochberg statistical procedures for threshold graph identification are distribution free in sign similarity network in the class of elliptically contoured distributions. Moreover it is shown that these procedures can be adapted for distribution free threshold graph identification in Pearson correlation network.

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.

Using network models to investigate the interconnectivity in modern economic systems allows researchers to better understand and explain some economic phenomena. This volume presents contributions by known experts and active researchers in economic and financial network modeling. Readers are provided with an understanding of the latest advances in network analysis as applied to economics, finance, corporate governance, and investments. Moreover, recent advances in market network analysis that focus on influential techniques for market graph analysis are also examined. Young researchers will find this volume particularly useful in facilitating their introduction to this new and fascinating field. Professionals in economics, financial management, various technologies, and network analysis, will find the network models presented in this book beneficial in analyzing the interconnectivity in modern economic systems.

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.

Full texts of third international conference on data analytics are presented.

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.