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.

Superconducting fault current limiter design using parallel-connected YBCO thin films

Superconducting Fault Current Limiters (SFCLs) are able to reduce fault currents to an acceptable value, reducing potential mechanical and thermal damage to power system apparatus and allowing more flexibility in power system design and operation. The device can also help avoid replacing circuit breakers whose capacity has been exceeded. Due to limitations in current YBCO thin film manufacturing processes, it is not easy to obtain one large thin film that satisfies the specifications for high voltage and large current applications. The combination of standardized thin films has merit to reduce costs and maintain device quality, and it is necessary to connect these thin films in different series and parallel configurations in order to meet these specifications. In this paper, the design of a resistive type SFCL using parallel-connected YBCO thin films is discussed, including the role of a parallel resistor and the influence of individual thin film characteristics, based on both theory and experimental results. © 2009 IEEE.

Spontaneous DC Current Generation in a Resistively Shunted Semiconductor Superlattice Driven by a TeraHertz Field

We study a resistively shunted semiconductor superlattice subject to a high-frequency electric field. Using a balance equation approach that incorporates the influence of the electric circuit, we determine numerically a range of amplitude and frequency of the ac field for which a dc bias and current are generated spontaneously and show that this region is likely accessible to current experiments. Our simulations reveal that the Bloch frequency corresponding to the spontaneous dc bias is approximately an integer multiple of the ac field frequency.

Surface determination of the air-earth electrical current density using co-located sensors of different geometry

A vertical conduction current flows in the atmosphere as a result of the global atmospheric electric circuit. The current at the surface consists of the conduction current and a locally generated displacement current, which are often approximately equal in magnitude. A method of separating the two currents using two collectors of different geometry is investigated. The picoammeters connected to the collectors have a RC time constant of approximately 3 s, permitting the investigation of higher frequency air-earth current changes than previously achieved. The displacement current component of the air-earth current derived from the instrument agrees with calculations using simultaneous data from a co-located fast response electric field mill. The mean value of the nondisplacement current measured over 9 h was 1.76 +/- 0.002 pA m(-2). (c) 2006 American Institute of Physics.

High performance blocking filters for the region 1µm to 20µm

A set of filters based on the sequence of semiconductor edges is described which offers continuity of short-wave infrared blocking. The rejection throughout the stop region is greater than 103 for each filter and the transmission better than 70% through one octave with a square cutoff. The cutoff points are located at intervals of about two-thirds of an octave. Filters at 2.6 ,µm, 5.5 µm, and 12 µm which use a low-passing multilayer in combination with a semiconductor absorption edge are described in detail. The design of multilayers for optimum performance is discussed by analogy with the synthesis of electric circuit filters.

Experimental determination of layer cloud edge charging from cosmic ray ionisation

The cloud-air transition zone at stratiform cloud edges is an electrically active region where droplet charging has been predicted. Cloud edge droplet charging is expected from vertical flow of cosmic ray generated atmospheric ions in the global electric circuit. Experimental confirmation of stratiform cloud edge electrification is presented here, through charge and droplet measurements made within an extensive layer of supercooled stratiform cloud, using a specially designed electrostatic sensor. Negative space charge up to 35 pC m−3 was found in a thin (<100 m) layer at the lower cloud boundary associated with the clear air-cloud conductivity gradient, agreeing closely with space charge predicted from the measured droplet concentration using ion-aerosol theory. Such charge levels carried by droplets are sufficient to influence collision processes between cloud droplets.

Extraction of Tschebysheff design data for the low pass dielectric multilayer

The extraction of design data for the lowpass dielectric multilayer according to Tschebysheff performance is described. The extraction proceeds initially by analogy with electric-circuit design, and can then be given numerical refinement which is also described. Agreement with the Tschebysheff desideratum is satisfactory. The multilayers extracted by this procedure are of fractional thickness, symmetric with regard to their central layers.

Another Proof of the Preceding Theory

Another Proof of the Preceding Theory was produced as part of a residency run by Artists in Archeology in conjunction with the Stonehenge Riverside project. The film explores the relationship between science, work and ritual, imagining archaeology as a future cult. As two robed disciples stray off from the dig, they are drawn to the drone of the stones and proceed to play the henge like a gigantic Theremin. Just as a Theremin is played with the hand interfering in an electric circuit and producing sound without contact, so the stones respond to the choreographed bodily proximity. Finally, one of the two continues alone to the avenue at Avebury, where the magnetic pull of the stones reaches its climax. Shot on VHS, the film features a score by Zuzushi Monkey, with percussion and theremin sounds mirroring the action. The performers are mostly artists and archeologists from the art and archaeology teams. The archeologists were encouraged to perform their normal work in the robes, in an attempt to explore the meeting points of science and ritual and interrogate our relationship to an ultimately unknowable prehistoric past where activities we do not understand are relegated to the realm of religion. Stonehenge has unique acoustic properties, it’s large sarsen stones are finely worked on the inside, left rough on the outside, intensifying sound waves within the inner horseshoe, but since their real use, having been built over centuries, remains ambiguous, the film proposes that our attempts to decode them may themselves become encoded in their cumulative meaning for future researchers.

Charge measurements in stratiform cloud from a balloon based sensor

Abstract. The electrification of stratiform clouds has is little investigated in comparison with thunderstorms and fair weather atmospheric electricity. Theory indicates that, at the upper and lower horizontal boundaries of layer clouds, charging will arise from vertical flow of cosmogenic ions in the global atmospheric electric circuit. Charge is transferred to droplets and particles, affecting cloud microphysical processes such as collision and droplet activation. Due to the lack of in-situ measurements, the magnitude and distribution of charge in stratiform clouds is not well known. A sensitive, inexpensive, balloon borne charge sensor has been developed to make in-situ measurements of edge charging in stratiform cloud using a standard meteorological radiosonde system. The charge sensor has now been flown through over 20 stratiform clouds and frequently detected charge up to 200 pC m-3 near cloud edges. These results are compared with measurements from the same sensor used to investigate charge in particle layers, such as volcanic ash from the Eyjafjallajökull eruption, and Saharan dust in the Cape Verde Isles. 1.

Cloud base height and cosmic rays

Cosmic rays modify current flow in the global atmospheric electrical circuit. Charging at horizontal layer cloud edges has been observed to be consistent with global circuit vertical current flow through the cloud, which can modify the properties of small and pure water droplets. Studies have been hampered by the absence of cloud edge observations, hence cloud base height information is investigated here. Cloud base height measured at the Lerwick Observatory, Shetland, UK, is analysed using threshold tests and spectral analysis. The cloud base height distributions for low cloud (cloud base less than 800 m) are found to vary with cosmic ray conditions. Further, 27 day and 1.68 year periodicities characteristic of cosmic ray variations are present, weakly, in the cloud base height data of stratiform clouds, when such periodicities are present in neutron monitor cosmic ray data. These features support the idea of propagation of heliospheric variability into layer clouds, through the global atmospheric electric circuit.

Inferring convective responses to El Niño with atmospheric electricity measurements at Shetland

Pacific ocean temperature anomalies associated with the El Niño–Southern Oscillation (ENSO) modulate atmospheric convection and hence thunderstorm electrification. The generated current flows globally via the atmospheric electric circuit, which can be monitored anywhere on Earth. Atmospheric electricity measurements made at Shetland (in Scotland) display a mean global circuit response to ENSO that is characterized by strengthening during 'El Niño' conditions, and weakening during 'La Niña' conditions. Examining the hourly varying response indicates that a potential gradient (PG) increase around noon UT is likely to be associated with a change in atmospheric convection and resultant lightning activity over equatorial Africa and Eastern Asia. A secondary increase in PG just after midnight UT can be attributed to more shower clouds in the central Pacific ocean during an 'El Niño'.

Measurements of atmospheric electricity aloft

Measurements of the electrical characteristics of the atmosphere above the surface have been made for over 200 years, from a variety of different platforms, including kites, balloons, rockets and aircraft. From these measurements, a great deal of information about the electrical characteristics of the atmosphere has been gained, assisting our understanding of the global atmospheric electric circuit, thunderstorm electrification and lightning generation mechanisms, discovery of transient luminous events above thunderstorms, and many other electrical phenomena. This paper surveys the history of atmospheric electrical measurements aloft, from the earliest manned balloon ascents to current day observations with free balloons and aircraft. Measurements of atmospheric electrical parameters in a range of meteorological conditions are described, including clear air conditions, polluted conditions, non-thunderstorm clouds, and thunderstorm clouds, spanning a range of atmospheric conditions, from fair weather, to the most electrically active.

Quantifying the importance of galactic cosmic rays in cloud microphysical processes

Galactic Cosmic Rays are one of the major sources of ion production in the troposphere and stratosphere. Recent studies have shown that ions form electrically charged clusters which may grow to become cloud droplets. Aerosol particles charge by the attachment of ions and electrons. The collision efficiency between a particle and a water droplet increases, if the particle is electrically charged, and thus aerosol-cloud interactions can be enhanced. Because these microphysical processes may change radiative properties of cloud and impact Earth's climate it is important to evaluate these processes' quantitative effects. Five different models developed independently have been coupled to investigate this. The first model estimates cloud height from dew point temperature and the temperature profile. The second model simulates the cloud droplet growth from aerosol particles using the cloud parcel concept. In the third model, the scavenging rate of the aerosol particles is calculated using the collision efficiency between charged particles and droplets. The fourth model calculates electric field and charge distribution on water droplets and aerosols within cloud. The fifth model simulates the global electric circuit (GEC), which computes the conductivity and ionic concentration in the atmosphere in altitude range 0–45 km. The first four models are initially coupled to calculate the height of cloud, boundary condition of cloud, followed by growth of droplets, charge distribution calculation on aerosols and cloud droplets and finally scavenging. These models are incorporated with the GEC model. The simulations are verified with experimental data of charged aerosol for various altitudes. Our calculations showed an effect of aerosol charging on the CCN concentration within the cloud, due to charging of aerosols increase the scavenging of particles in the size range 0.1 µm to 1 µm.

Simulation of a diffusion driven mobile robot

In this paper, we investigate the possibility to control a mobile robot via a sensory-motory coupling utilizing diffusion system. For this purpose, we implemented a simulation of the diffusion process of chemicals and the kinematics of the mobile robot. In comparison to the original Braitenberg vehicle in which sensorymotor coupling is tightly realised by hardwiring, our system employs the soft coupling. The mobile robot has two sets of independent sensory-motor unit, two sensors are implemented in front and two motors on each side of the robot. The framework used for the sensory-motor coupling was such that 1) Place two electrodes in the medium 2) Drop a certain amount of Chemical U and V related to the distance to the walls and the intensity of the light 3) Place other two electrodes in the medium 4) Measure the concentration of Chemical U and V to actuate the motors on both sides of the robot. The environment was constructed with four surrounding walls and a light source located at the center. Depending on the design parameters and initial conditions, the robot was able to successfully avoid the wall and light. More interestingly, the diffusion process in the sensory-motor coupling provided the robot with a simple form of memory which would not have been possible with a control framework based on a hard-wired electric circuit.

What do cosmogenic isotopes tell us about past solar forcing of climate?

In paleoclimate studies, cosmogenic isotopes are frequently used as proxy indicators of past variations in solar irradiance on centennial and millennial timescales. These isotopes are spallation products of galactic cosmic rays (GCRs) impacting Earth’s atmosphere, which are deposited and stored in terrestrial reservoirs such as ice sheets, ocean sediments and tree trunks. On timescales shorter than the variations in the geomagnetic field, they are modulated by the heliosphere and thus they are, strictly speaking, an index of heliospheric variability rather than one of solar variability. Strong evidence of climate variations associated with the production (as opposed to the deposition) of these isotopes is emerging. This raises a vital question: do cosmic rays have a direct influence on climate or are they a good proxy indicator for another factor that does (such as the total or spectral solar irradiance)? The former possibility raises further questions about the possible growth of air ions generated by cosmic rays into cloud condensation nuclei and/or the modulation of the global thunderstorm electric circuit. The latter possibility requires new understanding about the required relationship between the heliospheric magnetic fields that scatter cosmic rays and the photospheric magnetic fields which modulate solar irradiance.