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.

Magnetospheric access of solar particles and the configuration of the distant geomagnetic field

The access of 1.2-40 MeV protons and 0.4-1.0 MeV electrons from interplanetary space to the polar cap regions has been investigated with an experiment on board a low altitude, polar orbiting satellite (OG0-4).

A total of 333 quiet time observations of the electron polar cap boundary give a mapping of the boundary between open and closed geomagnetic field lines which is an order of magnitude more comprehensive than previously available.

Persistent features (north/south asymmetries) in the polar cap proton flux, which are established as normal during solar proton events, are shown to be associated with different flux levels on open geomagnetic field lines than on closed field lines. The pole in which these persistent features are observed is strongly correlated to the sector structure of the interplanetary magnetic field and uncorrelated to the north/south component of this field. The features were observed in the north (south) pole during a negative (positive) sector 91% of the time, while the solar field had a southward component only 54% of the time. In addition, changes in the north/south component have no observable effect on the persistent features.

Observations of events associated with co-rotating regions of enhanced proton flux in interplanetary space are used to establish the characteristics of the 1.2 - 40 MeV proton access windows: the access window for low polar latitudes is near the earth, that for one high polar latitude region is ~250 R_⊕ behind the earth, while that for the other high polar latitude region is ~1750 R_⊕ behind the earth. All of the access windows are of approximately the same extent (~120 R_⊕). The following phenomena contribute to persistent polar cap features: limited interplanetary regions of enhanced flux propagating past the earth, radial gradients in the interplanetary flux, and anisotropies in the interplanetary flux.

These results are compared to the particle access predictions of the distant geomagnetic tail configurations proposed by Michel and Dessler, Dungey, and Frank. The data are consistent with neither the model of Michel and Dessler nor that of Dungey. The model of Frank can yield a consistent access window configuration provided the following constraints are satisfied: the merging rate for open field lines at one polar neutral point must be ~5 times that at the other polar neutral point, related to the solar magnetic field configuration in a consistent fashion, the migration time for open field lines to move across the polar cap region must be the same in both poles, and the open field line merging rate at one of the polar neutral points must be at least as large as that required for almost all the open field lines to have merged in 0 (one hour). The possibility of satisfying these constraints is investigated in some detail.

The role played by interplanetary anisotropies in the observation of persistent polar cap features is discussed. Special emphasis is given to the problem of non-adiabatic particle entry through regions where the magnetic field is changing direction. The degree to which such particle entry can be assumed to be nearly adiabatic is related to the particle rigidity, the angle through which the field turns, and the rate at which the field changes direction; this relationship is established for the case of polar cap observations.