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.

Discovery of non-zero neutrino mixing angle θ_13 using Daya Bay antineutrino detectors

The Daya Bay Reactor Antineutrino Experiment observed the disappearance of reactor $\bar{\nu}_e$ from six $2.9~GW_{th}$ reactor cores in Daya Bay, China. The Experiment consists of six functionally identical $\bar{\nu}_e$ detectors, which detect $\bar{\nu}_e$ by inverse beta decay using a total of about 120 metric tons of Gd-loaded liquid scintillator as the target volume. These $\bar{\nu}_e$ detectors were installed in three underground experimental halls, two near halls and one far hall, under the mountains near Daya Bay, with overburdens of 250 m.w.e, 265 m.w.e and 860 m.w.e. and flux-weighted baselines of 470 m, 576 m and 1648 m. A total of 90179 $\bar{\nu}_e$ candidates were observed in the six detectors over a period of 55 days, 57549 at the Daya Bay near site, 22169 at the Ling Ao near site and 10461 at the far site. By performing a rate-only analysis, the value of $sin^2 2\theta_{13}$ was determined to be $0.092 \pm 0.017$.

Projections in a normed linear space and a generalization of the pseudo-inverse

The concept of a "projection function" in a finite-dimensional real or complex normed linear space H (the function P_M which carries every element into the closest element of a given subspace M) is set forth and examined.

If dim M = dim H - 1, then P_M is linear. If P_N is linear for all k-dimensional subspaces N, where 1 ≤ k < dim M, then P_M is linear.

The projective bound Q, defined to be the supremum of the operator norm of P_M for all subspaces, is in the range 1 ≤ Q < 2, and these limits are the best possible. For norms with Q = 1, P_M is always linear, and a characterization of those norms is given.

If H also has an inner product (defined independently of the norm), so that a dual norm can be defined, then when P_M is linear its adjoint P_M^H is the projection on (kernel P_M)^⊥ by the dual norm. The projective bounds of a norm and its dual are equal.

The notion of a pseudo-inverse F⁺ of a linear transformation F is extended to non-Euclidean norms. The distance from F to the set of linear transformations G of lower rank (in the sense of the operator norm ∥F - G∥) is c/∥F⁺∥, where c = 1 if the range of F fills its space, and 1 ≤ c < Q otherwise. The norms on both domain and range spaces have Q = 1 if and only if (F⁺)⁺ = F for every F. This condition is also sufficient to prove that we have (F⁺)^H = (F^H)⁺, where the latter pseudo-inverse is taken using dual norms.

In all results, the real and complex cases are handled in a completely parallel fashion.

Origin of Th/U variations in chondritic meteorites

Isotope dilution thorium and uranium analyses of the Harleton chondrite show a larger scatter than previously observed in equilibrated ordinary chondrites (EOC). The linear correlation of Th/U with 1/U in Harleton (and all EOC data) is produced by variation in the chlorapatite to merrillite mixing ratio. Apatite variations control the U concentrations. Phosphorus variations are compensated by inverse variations in U to preserve the Th/U vs. 1/U correlation. Because the Th/U variations reflect phosphate ampling, a weighted Th/U average should converge to an improved solar system Th/U. We obtain Th/U=3.53 (1_-mean=0.10), significantly lower and more precise than previous estimates.

To test whether apatite also produces Th/U variation in CI and CM chondrites, we performed P analyses on the solutions from leaching experiments of Orgueil and Murchison meteorites.

A linear Th/U vs. 1/U correlation in CI can be explained by redistribution of hexavalent U by aqueous fluids into carbonates and sulfates.

Unlike CI and EOC, whole rock Th/U variations in CMs are mostly due to Th variations. A Th/U vs. 1/U linear correlation suggested by previous data for CMs is not real. We distinguish 4 components responsible for the whole rock Th/U variations: (1) P and actinide-depleted matrix containing small amounts of U-rich carbonate/sulfate phases (similar to CIs); (2) CAIs and (3) chondrules are major reservoirs for actinides, (4) an easily leachable phase of high Th/U. likely carbonate produced by CAI alteration. Phosphates play a minor role as actinide and P carrier phases in CM chondrites.

Using our Th/U and minimum galactic ages from halo globular clusters, we calculate relative supernovae production rates for ²³²Th/²³⁸U and ²³⁵U/²³⁸U for different models of r-process nucleosynthesis. For uniform galactic production, the beginning of the r-process nucleosynthesis must be less than 13 Gyr. Exponentially decreasing production is also consistent with a 13 Gyr age, but very slow decay times are required (less than 35 Gyr), approaching the uniform production. The 15 Gyr Galaxy requires either a fast initial production growth (infall time constant less than 0.5 Gyr) followed by very low decrease (decay time constant greater than 100 Gyr), or the fastest possible decrease (≈8 Gyr) preceded by slow in fall (≈7.5 Gyr).

Probability in Quantum Mechanics: The Everett Interpretation and the Decision-Theoretic Program

The Everett interpretation of quantum mechanics is an increasingly popular alternative to the traditional Copenhagen interpretation, but there are a few major issues that prevent the widespread adoption. One of these issues is the origin of probabilities in the Everett interpretation, which this thesis will attempt to survey. The most successful resolution of the probability problem thus far is the decision-theoretic program, which attempts to frame probabilities as outcomes of rational decision making. This marks a departure from orthodox interpretations of probabilities in the physical sciences, where probabilities are thought to be objective, stemming from symmetry considerations. This thesis will attempt to offer evaluations on the decision-theoretic program.

Plasma inverse scattering theory

The object of this report is to calculate the electron density profile of plane stratified inhomogeneous plasmas. The electron density profile is obtained through a numerical solution of the inverse scattering algorithm.

The inverse scattering algorithm connects the time dependent reflected field resulting from a δ-function field incident normally on the plasma to the inhomogeneous plasma density.

Examples show that the method produces uniquely the electron density on or behind maxima of the plasma frequency.

It is shown that the δ-function incident field used in the inverse scattering algorithm can be replaced by a thin square pulse.