### Article

## Equivalence of Network Structures in Networks of Random Variables with Known and Unknown Shift Parameter

It is proved that sign network with elliptical distribution with known shift parameter is equivalent

to sign network with elliptical distribution with unknown shift parameter estimated as sample mean.

This result is proved for the case of independent identically distributed observation and for the case

of sample from matrix elliptically contoured distribution with any dependence between observations.

Invariance properties of statistical procedures for threshold graph identification are considered. An optimal procedure in the class of invariant multiple decision procedures is constructed.

Market network analysis attracts a growing attention last decades. One of the most important problems related with it is the detection of dynamics in market network. In the present paper, the stock market network of stock’s returns is considered. Probability of sign coincidence of stock’s returns is used as the measure of similarity between stocks. Robust (distribution free) multiple testing statistical procedure for testing dynamics of network is proposed. The constructed procedure is applied for German, French, UK, and USA market. It is shown that in most cases where the dynamics is observed it is determined by a small number of hubs in the associated rejection graph.

In each node of a network, economy is described by the simple two-period Romer’s model of endogenous growth with production and knowledge externalities. The sum of knowledge levels in the neighbor nodes causes an externality in the production of each node of the network. The game equilibrium in the network is investigated. The agents’ solutions depending on the size of externality are obtained. The uniqueness of inner equilibrium is proved. The role of passive agents in network formation is studied; in particular, the possibilities of adding a passive agent to a regular network, and also of joining of regular networks through nodes with passive agents. It is shown that the sum of knowledge levels in all the nodes decreases under adding of a new link.

Identification of network structures using the finite-size sample has been considered.

The concepts of random variables network and network model, which is a complete weighted

graph, have been introduced. Two types of network structures have been investigated: network

structures with an arbitrary number of elements and network structures with a fixed number

of elements of the network model. The problem of identification of network structures has

been investigated as a multiple testing problem. The risk function of statistical procedures for

identification of network structures can be represented as a linear combination of expected

numbers of incorrectly included elements and incorrectly non-included elements. The sufficient

conditions of optimality for statistical procedures for network structures identification with

an arbitrary number of elements have been given. The concept of statistical uncertainty of

statistical procedures for identification of network structures has been introduced.

This paper considers a network game as follows. In each node of a network, economy is described by the simple two-period Romer’s model of endogenous growth with production and knowledge externalities. The sum of knowledge levels in the neighbor nodes causes an externality in the production of each network node. The concept of node type is introduced and a corresponding typology of networks is suggested. As demonstrated below, all inner equilibria of the game are determined by this typology. For several typologies, the equilibrium knowledge levels are found in explicit form for the nodes that have different positions in the network.

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.

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.