Abstract:
Networks are useful for describing systems of interacting objects, where the nodes represent the objects and the edges represent the interactions between them. The applications include chemical and metabolic systems, food webs as well as social networks. Lately, it was found that many of these networks display some common topological features, such as high clustering, small average path length (small world networks) and a power-law degree distribution (scale free networks). The topological features of a network are commonly related to the network's functionality. However, the topology alone does not account for the nature of the interactions in the network and their strength. Here we introduce a method for evaluating the correlations between pairs of nodes in the network. These correlations depend both on the topology and on the functionality of the network. A network with high connectivity displays strong correlations between its interacting nodes and thus features small-world functionality. We quantify the correlations between all pairs of nodes in the network, and express them as matrix elements in the correlation matrix. From this information one can plot the correlation function for the network and to extract the correlation length. The connectivity of a network is then defined as the ratio between this correlation length and the average path length of the network. Using this method we distinguish between a topological small world and a functional small world, where the latter is characterized by long range correlations and high connectivity. Clearly, networks which share the same topology, may have different connectivities, based on the nature and strength of their interactions. The method is demonstrated on metabolic networks, but can be readily generalized to other types of networks.

Abstract:
This thesis is a compendium of research which brings together ideas from the fields of Complex Networks and Computational Neuroscience to address two questions regarding neural systems: 1) How the activity of neurons, via synaptic changes, can shape the topology of the network they form part of, and 2) How the resulting network structure, in its turn, might condition aspects of brain behaviour. Although the emphasis is on neural networks, several theoretical findings which are relevant for complex networks in general are presented -- such as a method for studying network evolution as a stochastic process, or a theory that allows for ensembles of correlated networks, and sets of dynamical elements thereon, to be treated mathematically and computationally in a model-independent manner. Some of the results are used to explain experimental data -- certain properties of brain tissue, the spontaneous emergence of correlations in all kinds of networks... -- and predictions regarding statistical aspects of the central nervous system are made. The mechanism of Cluster Reverberation is proposed to account for the near-instant storage of novel information the brain is capable of.

Abstract:
Networks are becoming a ubiquitous metaphor for the understanding of complex biological systems, spanning the range between molecular signalling pathways, neural networks in the brain, and interacting species in a food web. In many models, we face an intricate interplay between the topology of the network and the dynamics of the system, which is generally very hard to disentangle. A dynamical feature that has been subject of intense research in various fields are correlations between the noisy activity of nodes in a network. We consider a class of systems, where discrete signals are sent along the links of the network. Such systems are of particular relevance in neuroscience, because they provide models for networks of neurons that use action potentials for communication. We study correlations in dynamic networks with arbitrary topology, assuming linear pulse coupling. With our novel approach, we are able to understand in detail how specific structural motifs affect pairwise correlations. Based on a power series decomposition of the covariance matrix, we describe the conditions under which very indirect interactions will have a pronounced effect on correlations and population dynamics. In random networks, we find that indirect interactions may lead to a broad distribution of activation levels with low average but highly variable correlations. This phenomenon is even more pronounced in networks with distance dependent connectivity. In contrast, networks with highly connected hubs or patchy connections often exhibit strong average correlations. Our results are particularly relevant in view of new experimental techniques that enable the parallel recording of spiking activity from a large number of neurons, an appropriate interpretation of which is hampered by the currently limited understanding of structure-dynamics relations in complex networks.

Abstract:
Network topology optimization has been widely researched. Since market competition has gradually developed into competition among the supply chain information systems, the network topology optimization of supply chain information systems has been in urgent need. However, the network topology optimization of supply chain information systems is still in its early stages and still has some challenges. So a description of typical seven network topologies for various supply chain information systems has been given. The generic characteristics of each network topology can be summarized. To analyze the optimization of network topology optimization of supply chain information systems, a numeric model has been established based on these general characteristics. A genetic algorithm is applied in the network topology optimization of supply chain information systems model to achieve the minimum cost and shortest path. Finally, our experiment results are provided to demonstrate the robustness and effectiveness of the proposed model. Network topology optimization has been widely researched. Since market competition has gradually developed into competition among the supply chain information systems, the network topology optimization of supply chain information systems has been in urgent need. However, the network topology optimization of supply chain information systems is still in its early stages and still has some challenges. So a description of typical seven network topologies for various supply chain information systems has been given. The generic characteristics of each network topology can be summarized. To analyze the optimization of network topology optimization of supply chain information systems, a numeric model has been established based on these general characteristics. A genetic algorithm is applied in the network topology optimization of supply chain information systems model to achieve the minimum cost and shortest path. Finally, our experiment results are provided to demonstrate the robustness and effectiveness of the proposed model.

Abstract:
Theoretical basis for the functioning of the software package intended for the design and investigation of communication systems with a dynamic network topology are developed in the paper. A mathematical model for assessing the ef-ficiency of designed telecommunication networks is developed. Necessary and sufficient conditions needed to ensure the connectivity of communication sys-tems with a dynamic network topology are formulated. An algorithm of the software package operation for assessing the quality of functioning of telecommunication systems with a dynamic network topology is developed.

Abstract:
The network modernization, educational information systems software and hardware updates problem is actual in modern term of information technologies prompt development. There are server applications and network topology of Institute of Information Technology and Learning Tools of National Academy of Pedagogical Sciences of Ukraine analysis and their improvement methods expound in the article. The article materials represent modernization results implemented to increase network efficiency and reliability, decrease response time in Institute’s network information systems. The article gives diagrams of network topology before upgrading and after finish of optimization and upgrading processes. Модерн зац я мереж, оновлення програмного та апаратного забезпечення нформац йних систем у галуз осв ти актуальною проблемою в сучасних умовах швидкого розвитку нформац йних технолог й. У статт подано анал з серверного забезпечення та мережево тополог нтернет-центру нституту нформац йних технолог й засоб в навчання НАПН Укра ни, запропоновано шляхи модерн зац х. Матер али статт в дображають результати роботи з модерн зац мереж , проведен з метою п двищення ефективност й над йност мереж , скорочення часу на оч кування результат в даних обробки в нформац йних системах нституту п д час роботи в мереж . У статт наведено схеми мережево тополог до модерн зац та п сля проведення.

Abstract:
Systems of networked mobile robots, such as unmanned aerial or ground vehicles, will play important roles in future military and commercial applications. The communications for such systems will typically be over wireless links and may require that the robots form an ad hoc network and communicate on a peer-to-peer basis. In this paper, we consider the problem of optimizing the network topology to minimize the total traffic in a network required to support a given set of data flows under constraints on the amount of movement possible at each mobile robot. In this paper, we consider a subclass of this problem in which the initial and final topologies are trees, and the movement restrictions are given in terms of the number of edges in the graph that must be traversed. We develop algorithms to optimize the network topology while maintaining network connectivity during the topology reconfiguration process. Our topology reconfiguration algorithm uses the concept of prefix labelling and routing to move nodes through the network while maintaining network connectivity. We develop two algorithms to determine the final network topology: an optimal, but computationally complex algorithm, and a greedy suboptimal algorithm that has much lower complexity. We present simulation results to compare the performance of these algorithm.

Abstract:
We study spatial correlations in the transport of energy between two baths at different temperatures. To do this, we introduce a minimal model in which energy flows from one bath to another through two subsystems. We show that the transport-induced energy correlations between the two subsystems are of the same order as the energy fluctuations within each subsystem. The correlations can be either positive or negative and we give bounds on their values which are associated with a dynamic energy scale. The different signs originate as a competition between fluctuations generated near the baths, and fluctuations of the current between the two subsystems. This interpretation sheds light on known results for spatially-dependent heat and particle conduction models.

Abstract:
We introduce an extension of the dynamical mean field approximation (DMFA) which retains the causal properties and generality of the DMFA, but allows for systematic inclusion of non-local corrections. Our technique maps the problem to a self-consistently embedded cluster. The DMFA (exact result) is recovered as the cluster size goes to one (infinity). As a demonstration, we study the Falicov-Kimball model using a variety of cluster sizes. We show that the sum rules are preserved, the spectra are positive definite, and the non-local correlations suppress the CDW transition temperature. \

Abstract:
Low-dimensional systems are an important field of current theoretical and experimental research. Recent technological developments provide many possible realizations of effectively one-dimensional systems. These devices promise to give us access to a new range of phenomena. It is therefore very interesting to develop theoretical methods specific for such systems to model their behavior and calculate the correlators of the resulting theory. Incidentally, one such method exists and is known as Bosonization. It can be applied to one-dimensional systems and effectively describes low energy excitations in a universal way. We use the example of a correlator known as the Emptiness Formation Probability to show that Bosonization fails to describe some long range correlators corresponding to large disturbances (the EFP measures the probability for the ground state of the system to develop a region without particles). We trace this failure to the fact that Bosonization is constructed as a linear approximation of the full theory and we set up to develop a collective description with the required non-linearity. The resulting scheme is essentially a Hydrodynamic paradigm for quantum systems. We show how to construct such a hydrodynamic description for a variety of exactly integrable models and illustrate how it can be used to make new predictions. For the special case of the spin-1/2 XY model we take advantage of the structure of the model to express the EFP as a determinant of a very special type of matrix, known as Toeplitz Matrix. We use the theory of Toeplitz determinants to calculate the asymptotic behavior of the EFP in the XY model and discuss its relation with the criticality of the theory.