### Book

## Network Models in Economics and Finance

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.

Two classes of two- and three-person games on polyhedral sets of player strategies that appear in estimating fair shares of the market participants in a marketplace are considered. In games from both classes, payoff functions of the players are sums of linear functions of vector arguments or those of linear ones and a bilinear function. Games from the first class are those in which player strategies are connected, i.e., they cannot be chosen by the players independently, whereas player strategies in games from the second class are disjoint. For the games from both classes either sucient or necessary and sucient conditions of the equilibriums are provided, and these conditions allow one to calculate the equilibriums by effective optimization techniques. This fact contributes to making the equilibrium concept a productive approach to quantitatively analyzing conflicts in systems economic studies. Economic problems that appear in systems described by nonlinear mathematical models with linear constraints, in particular, by some network models, including a) restructuring a company and positioning the restructured company in a market or in several markets, b) forming a pool of regional clients interested in selling their products and in buying somebody else's ones outside their regions via forward contracts offered by regional brokers, c) determining initial prices for procurement contracts to be tendered by a public administration, d) nding competitive transportation taris by a railroad company competing with tracking companies for providing transportation services both in a region and between two regions, e) calculating an optimal volume of producing electricity by a base load power plant in a part of a country's electrical grid under an uncertain demand in the corresponding network of the grid customers, and f) forming a public-private partnership to develop a set of projects that a public administration needs to develop and implement, but does not have funds to nance on its own (partly or even completely) are discussed to illustrate how the games under consideration appear, and how they can be analyzed.

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.

Current approaches to testing hypotheses on degree distribution of the market graph and of identifying power law in data are discussed, main drawbacks of these approaches are identified and ways to overcome them are provided. Brief summary on methodology used is given and main aspects are highlighted. Discussed methodology is applied to testing hypotheses on degree distribution of multiple market graphs and results obtained are presented. It is shown that more stable market research techniques question presence of power law in degree distribution.

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

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.

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

A model for organizing cargo transportation between two node stations connected by a railway line which contains a certain number of intermediate stations is considered. The movement of cargo is in one direction. Such a situation may occur, for example, if one of the node stations is located in a region which produce raw material for manufacturing industry located in another region, and there is another node station. The organization of freight traﬃc is performed by means of a number of technologies. These technologies determine the rules for taking on cargo at the initial node station, the rules of interaction between neighboring stations, as well as the rule of distribution of cargo to the ﬁnal node stations. The process of cargo transportation is followed by the set rule of control. For such a model, one must determine possible modes of cargo transportation and describe their properties. This model is described by a ﬁnite-dimensional system of diﬀerential equations with nonlocal linear restrictions. The class of the solution satisfying nonlocal linear restrictions is extremely narrow. It results in the need for the “correct” extension of solutions of a system of diﬀerential equations to a class of quasi-solutions having the distinctive feature of gaps in a countable number of points. It was possible numerically using the Runge–Kutta method of the fourth order to build these quasi-solutions and determine their rate of growth. Let us note that in the technical plan the main complexity consisted in obtaining quasi-solutions satisfying the nonlocal linear restrictions. Furthermore, we investigated the dependence of quasi-solutions and, in particular, sizes of gaps (jumps) of solutions on a number of parameters of the model characterizing a rule of control, technologies for transportation of cargo and intensity of giving of cargo on a node station.

Let k be a field of characteristic zero, let G be a connected reductive algebraic group over k and let g be its Lie algebra. Let k(G), respectively, k(g), be the field of k- rational functions on G, respectively, g. The conjugation action of G on itself induces the adjoint action of G on g. We investigate the question whether or not the field extensions k(G)/k(G)^G and k(g)/k(g)^G are purely transcendental. We show that the answer is the same for k(G)/k(G)^G and k(g)/k(g)^G, and reduce the problem to the case where G is simple. For simple groups we show that the answer is positive if G is split of type A_n or C_n, and negative for groups of other types, except possibly G_2. A key ingredient in the proof of the negative result is a recent formula for the unramified Brauer group of a homogeneous space with connected stabilizers. As a byproduct of our investigation we give an affirmative answer to a question of Grothendieck about the existence of a rational section of the categorical quotient morphism for the conjugating action of G on itself.

Let G be a connected semisimple algebraic group over an algebraically closed field k. In 1965 Steinberg proved that if G is simply connected, then in G there exists a closed irreducible cross-section of the set of closures of regular conjugacy classes. We prove that in arbitrary G such a cross-section exists if and only if the universal covering isogeny Ĝ → G is bijective; this answers Grothendieck's question cited in the epigraph. In particular, for char k = 0, the converse to Steinberg's theorem holds. The existence of a cross-section in G implies, at least for char k = 0, that the algebra k[G]G of class functions on G is generated by rk G elements. We describe, for arbitrary G, a minimal generating set of k[G]G and that of the representation ring of G and answer two Grothendieck's questions on constructing generating sets of k[G]G. We prove the existence of a rational (i.e., local) section of the quotient morphism for arbitrary G and the existence of a rational cross-section in G (for char k = 0, this has been proved earlier); this answers the other question cited in the epigraph. We also prove that the existence of a rational section is equivalent to the existence of a rational W-equivariant map T- - - >G/T where T is a maximal torus of G and W the Weyl group.