Mathematical study of complex networks : brain, internet, and power grid

The dissertation is concerned with the mathematical study of various network problems. First, three real-world networks are considered: (i) the human brain network (ii) communication networks, (iii) electric power networks. Although these networks perform very different tasks, they share similar mathematical foundations. The high-level goal is to analyze and/or synthesis each of these systems from a “control and optimization” point of view. After studying these three real-world networks, two abstract network problems are also explored, which are motivated by power systems. The first one is “flow optimization over a flow network” and the second one is “nonlinear optimization over a generalized weighted graph”. The results derived in this dissertation are summarized below.

Brain Networks: Neuroimaging data reveals the coordinated activity of spatially distinct brain regions, which may be represented mathematically as a network of nodes (brain regions) and links (interdependencies). To obtain the brain connectivity network, the graphs associated with the correlation matrix and the inverse covariance matrix—describing marginal and conditional dependencies between brain regions—have been proposed in the literature. A question arises as to whether any of these graphs provides useful information about the brain connectivity. Due to the electrical properties of the brain, this problem will be investigated in the context of electrical circuits. First, we consider an electric circuit model and show that the inverse covariance matrix of the node voltages reveals the topology of the circuit. Second, we study the problem of finding the topology of the circuit based on only measurement. In this case, by assuming that the circuit is hidden inside a black box and only the nodal signals are available for measurement, the aim is to find the topology of the circuit when a limited number of samples are available. For this purpose, we deploy the graphical lasso technique to estimate a sparse inverse covariance matrix. It is shown that the graphical lasso may find most of the circuit topology if the exact covariance matrix is well-conditioned. However, it may fail to work well when this matrix is ill-conditioned. To deal with ill-conditioned matrices, we propose a small modification to the graphical lasso algorithm and demonstrate its performance. Finally, the technique developed in this work will be applied to the resting-state fMRI data of a number of healthy subjects.

Communication Networks: Congestion control techniques aim to adjust the transmission rates of competing users in the Internet in such a way that the network resources are shared efficiently. Despite the progress in the analysis and synthesis of the Internet congestion control, almost all existing fluid models of congestion control assume that every link in the path of a flow observes the original source rate. To address this issue, a more accurate model is derived in this work for the behavior of the network under an arbitrary congestion controller, which takes into account of the effect of buffering (queueing) on data flows. Using this model, it is proved that the well-known Internet congestion control algorithms may no longer be stable for the common pricing schemes, unless a sufficient condition is satisfied. It is also shown that these algorithms are guaranteed to be stable if a new pricing mechanism is used.

Electrical Power Networks: Optimal power flow (OPF) has been one of the most studied problems for power systems since its introduction by Carpentier in 1962. This problem is concerned with finding an optimal operating point of a power network minimizing the total power generation cost subject to network and physical constraints. It is well known that OPF is computationally hard to solve due to the nonlinear interrelation among the optimization variables. The objective is to identify a large class of networks over which every OPF problem can be solved in polynomial time. To this end, a convex relaxation is proposed, which solves the OPF problem exactly for every radial network and every meshed network with a sufficient number of phase shifters, provided power over-delivery is allowed. The concept of “power over-delivery” is equivalent to relaxing the power balance equations to inequality constraints.

Flow Networks: In this part of the dissertation, the minimum-cost flow problem over an arbitrary flow network is considered. In this problem, each node is associated with some possibly unknown injection, each line has two unknown flows at its ends related to each other via a nonlinear function, and all injections and flows need to satisfy certain box constraints. This problem, named generalized network flow (GNF), is highly non-convex due to its nonlinear equality constraints. Under the assumption of monotonicity and convexity of the flow and cost functions, a convex relaxation is proposed, which always finds the optimal injections. A primary application of this work is in the OPF problem. The results of this work on GNF prove that the relaxation on power balance equations (i.e., load over-delivery) is not needed in practice under a very mild angle assumption.

Generalized Weighted Graphs: Motivated by power optimizations, this part aims to find a global optimization technique for a nonlinear optimization defined over a generalized weighted graph. Every edge of this type of graph is associated with a weight set corresponding to the known parameters of the optimization (e.g., the coefficients). The motivation behind this problem is to investigate how the (hidden) structure of a given real/complex valued optimization makes the problem easy to solve, and indeed the generalized weighted graph is introduced to capture the structure of an optimization. Various sufficient conditions are derived, which relate the polynomial-time solvability of different classes of optimization problems to weak properties of the generalized weighted graph such as its topology and the sign definiteness of its weight sets. As an application, it is proved that a broad class of real and complex optimizations over power networks are polynomial-time solvable due to the passivity of transmission lines and transformers.

Designing superresolution optical pupil filter with constrained global optimization algorithm

In this paper, a new method for designing three-zone optical pupil filter is presented. The phase-only optical pupil filter and the amplitude-only optical pupil filters were designed. The first kind of pupil for optical data storage can increase the transverse resolution. The second kind of pupil filter can increase the axial and transverse resolution at the same time, which is applicable in three-dimension imaging in confocal microscopy. (C) 2007 Elsevier GmbH. All rights reserved.

"Global Change" und das Meer als Nahrungsquelle

The potential importance of marine produetion as a protein ressource for a growing human population can hardly be overestimated. Climatic changes in the marine environment may affect marine production in a significant way. Increasing levels of UV-B may decrease primary production and thus diminish the food base for harvestable marine ressources. Direct effects on early stages of fishes may occur. Temperature changes can lead to additional mortality in the early phase of life histories of fishes. In spite of the potentially negative scenario, actual effects of global change on the ressources have not been detected so far. The marine organisms dispose of a significant level of pre-adaptation to changes of environmental factors both on a seasonal and an interannual scale. Effects on marine life may therefore be less dramatic than those on terrestrial systems, which are more directly linked with the exponentially growing human population.

Cost effectiveness of a combination of instruments for global warming: a quantitative approach for Spain

4 p.

The basque ecological debt: global socioecological impacts of a small open economy

4 p.

Linking Reduced Deforestation and a Global Carbon Market: Impacts on Costs, Financial Flows, and Technological Innovation

25 p.

Local and Global Externalities, Environmental Policies and Growth

33 p.

Local air pollution and global climate change taxes: a distributional analysis

25 p.

The influence of global warming on freshwater plankton communities in Britain

The likely response of freshwater plankton to the direct and indirect effects of sustained global warming are summarized. The increase in CO2 posited by climatologists will have a direct effect on many biological processes, and an even more important indirect effect on the global climate. Lake plankton populations are relatively well buffered against sudden fluctuations in temperature but can react in unexpected ways to seasonal changes in the wind speed, with effects on seasonal growth and succession of plankton. The direct

Feira de São Cristóvão: uma incursão sobre os estudos na área e a tentativa de aproximação à relação global-local

O presente trabalho se propõe fazer uma aproximação aos estudos sobre a tradicionalmente chamada Feira de São Cristóvão, levando em consideração a relação entre os agentes partícipes e a dinâmica no espaço. O objetivo é compreender melhor a força que move os atores no sentido da mudança da forma da feira para a forma do Centro Luiz Gonzaga de Tradições Nordestinas (CLGTN), e assim me aproximar da seguinte questão: Por que uma forma de comércio como a modelada na Feira de São Cristóvão, popularmente conhecida como feira dos paraíbas, sobrevive à força das mudanças operadas no ramo, dando lugar aos sofistificados shopping-centers? Para isso, faço uma aproximação ao estado de arte da pesquisa na área com base nas seguintes categorias de análise: a Feira como forma de produção do espaço e a relação Forma-Conteúdo, a relação espaço-identidade na feira e a relação entre agentes internos e externos no Espaço. Vendo a migração como movimento responsável pela constituição de um espaço social e territorialmente diferenciado do nordestino na cidade do Rio de Janeiro, é por ela que inicio a exposição. Esta se continua com a análise dos trabalhos escolhidos para estudo, via as categorias indicadas e, com um capítulo no qual procuro repensar a feira a partir da relação global-local que vem referenciando as reflexões acumuladas na área do urbano a partir da segunda metade do século XX. Por fim, concluo observando que a permanência da feira se deve a um duplo movimento de resistência-absorção que, se redefinindo no tempo e no espaço, não consegue, entretanto, apagar os traços da tradição que lhe deu origem.