Academic literature on the topic 'Complex conductance networks'

We study transport properties such as conductance and diffusion of complex networks such as scale-free and Erdős-Rényi networks. We consider the equivalent conductance G between two arbitrarily chosen nodes of random scale-free networks with degree distribution P(k) ~ k−⋋ and Erdős-Rényi networks in which each link has the same unit resistance. Our theoretical analysis for scale-free networks predicts a broad range of values of G (or the related diffusion constant D), with a power-law tail distribution ɸSF(G) ~ G−gG, where gG = 2⋋ − 1. We confirm our predictions by simulations of scale-free networks solving the Kirchhoff equations for the conductance between a pair of nodes. The power-law tail in ɸSF(G) leads to large values of G, thereby significantly improving the transport in scale-free networks, compared to Erdős-R´nyi networks where the tail of the conductivity distribution decays exponentially. Based on a simple physical “transport backbone” picture we suggest that the conductances of scale-free and Erdős-Rényi networks can be approximated by ckAkB/(kA + kB) for any pair of nodes A and B with degrees kA and kB. Thus, a single parameter c characterizes transport on both scale-free and Erdős-Rényi networks.

Doctor of Philosophy<br>Department of Electrical and Computer Engineering<br>Caterina Scoglio<br>Critical infrastructures are repeatedly attacked by external triggers causing tremendous amount of damages. Any infrastructure can be studied using the powerful theory of complex networks. A complex network is composed of extremely large number of different elements that exchange commodities providing significant services. The main functions of complex networks can be damaged by different types of attacks and failures that degrade the network performance. These attacks and failures are considered as disturbing dynamics, such as the spread of viruses in computer networks, the spread of epidemics in social networks, and the cascading failures in power grids. Depending on the network structure and the attack strength, every network differently suffers damages and performance degradation. Hence, quantifying the robustness of complex networks becomes an essential task. In this dissertation, new metrics are introduced to measure the robustness of technological and social networks with respect to the spread of epidemics, and the robustness of power grids with respect to cascading failures. First, we introduce a new metric called the Viral Conductance ($VC_{SIS}$) to assess the robustness of networks with respect to the spread of epidemics that are modeled through the susceptible/infected/susceptible ($SIS$) epidemic approach. In contrast to assessing the robustness of networks based on a classical metric, the epidemic threshold, the new metric integrates the fraction of infected nodes at steady state for all possible effective infection strengths. Through examples, $VC_{SIS}$ provides more insights about the robustness of networks than the epidemic threshold. In addition, both the paradoxical robustness of Barab\'si-Albert preferential attachment networks and the effect of the topology on the steady state infection are studied, to show the importance of quantifying the robustness of networks. Second, a new metric $VC_$ is introduced to assess the robustness of networks with respect to the spread of susceptible/infected/recovered ($SIR$) epidemics. To compute $VC_$, we propose a novel individual-based approach to model the spread of $SIR$ epidemics in networks, which captures the infection size for a given effective infection rate. Thus, $VC_$ quantitatively integrates the infection strength with the corresponding infection size. To optimize the $VC_$ metric, a new mitigation strategy is proposed, based on a temporary reduction of contacts in social networks. The social contact network is modeled as a weighted graph that describes the frequency of contacts among the individuals. Thus, we consider the spread of an epidemic as a dynamical system, and the total number of infection cases as the state of the system, while the weight reduction in the social network is the controller variable leading to slow/reduce the spread of epidemics. Using optimal control theory, the obtained solution represents an optimal adaptive weighted network defined over a finite time interval. Moreover, given the high complexity of the optimization problem, we propose two heuristics to find the near optimal solutions by reducing the contacts among the individuals in a decentralized way. Finally, the cascading failures that can take place in power grids and have recently caused several blackouts are studied. We propose a new metric to assess the robustness of the power grid with respect to the cascading failures. The power grid topology is modeled as a network, which consists of nodes and links representing power substations and transmission lines, respectively. We also propose an optimal islanding strategy to protect the power grid when a cascading failure event takes place in the grid. The robustness metrics are numerically evaluated using real and synthetic networks to quantify their robustness with respect to disturbing dynamics. We show that the proposed metrics outperform the classical metrics in quantifying the robustness of networks and the efficiency of the mitigation strategies. In summary, our work advances the network science field in assessing the robustness of complex networks with respect to various disturbing dynamics.

Thesis (Ph.D)--Mathematics, Georgia Institute of Technology, 2009.<br>Committee Chair: Mihail, Milena; Committee Member: Lu, Linyuan; Committee Member: Sokol, Joel; Committee Member: Tetali, Prasad; Committee Member: Trotter, Tom; Committee Member: Yu, Xingxing. Part of the SMARTech Electronic Thesis and Dissertation Collection.

Les argiles sont répandues dans la proche surface de la Terre, et ont un fort impact sur la perméabilité des formations géologiques. Leur très faible perméabilité fait des formations argileuses des "pièges géologiques" d’intérêt dans divers domaines d’étude des géosciences (notamment pour le pétrole et le gaz, la géothermie, le stockage des déchets nucléaires, entre autres). Les minéraux argileux présentent une charge de surface et une surface spécifique très importantes, ce qui génère le développement d’une double couche électrique particulièrement importante. La polarisation provoquée spectrale (PPS) est une méthode géoélectrique active qui permet d’obtenir de manière non-invasive la conductivité électrique complexe en fonction de la fréquence d’un géo-matériau du mHz au kHz. La conductivité complexe informe sur la capacité du matériau sondé à conduire un courant électrique et sur sa capacité à se polariser (à mobiliser de manière réversible des charges électriques). Cette thèse présente un protocole de laboratoire détaillé pour obtenir des mesures de PPS sur différents types d’argiles à des salinités variables, ainsi que des mélanges hétérogènes artificiels d’illite et demontmorillonite. Les résultats de la première étude montrent que la partie réelle de la conductivité électrique augmente avec la salinité, mais la partie imaginaire augmente jusqu’à un maximum et puis diminue. Cette diminution est due à la coagulation des argiles à hautes salinités. Cette coagulation potentielle des argiles altérerait l’espace poral puis modifierait les mécanismes de polarisation en jeu. Par ailleurs, en comparant le rapport de la conductivité de surface (imaginaire versus réelle) et d’autres données de la littérature, on remarque que ce rapport diminue avec la teneur en argile. Pour la deuxième étude, on observe que la montmorillonite domine la polarisation par rapport à l’illite. Cependant, les deux argiles ont un effet sur la conduction des mélanges. Les lois de mélanges sont une approche efficace pour modéliser ce type de mélange hétérogène d’argiles. Les modèles de réseaux de conductance complexes sont également utiles pour prédire la forme des spectres de polarisation. Les résultats de ce travail de thèse ouvrent de nouvelles perspectives pour la caractérisation des matériaux argileux avec la PPS<br>Clays are ubiquitously present in the Earth’s near surface and they have a high impact on the permeability of a system. Due to this property, clay formations are used in a variety of geology related applications (oil and gas, geothermal, nuclear waste storage, critical zone research, among others). Clays have a high surface charge and a high specific surface area, this property gives clays a particularly strong electrical double layer (EDL). Spectral induced polarization (SIP) is an active geo-electrical method thatmeasures in a non-invasive manner the frequency-dependent complex conductivity of a geo-material from themHz to the kHz. The complex conductivity informs about the ability the probed material has to conduct an electrical current and the ability to polarize (to reversibly store electrical charges). This thesis presents a detailed laboratory protocol to obtain SIP measurements of different types of clay at varying salinities, as well as an artificial heterogeneous mixture of illite and red montmorillonite with a salinity of around 10¡2 mol L¡1. The results of the first study show that the real part of the electrical conductivity increases with salinity, but the imaginary part increases until a maxima and then decreases. An interpretation of the decrease can come fromthe fact that clays coagulate at high salinities. The potential coagulation of clays would alter the pore space and then alter the polarization mechanisms in play. Furthermore, when comparing the ratio of the surface conductivity (imaginary versus real) of these resultswith other data in the literature, we notice that this ratio decreaseswith clay content. For the second study, we observe that red montmorillonite dominates the polarization with respect to illite. However, both clays effect the conduction of the mixtures. Mixing laws are an effective approach to model the complex conductivity of these heterogeneous mixtures. Complex conductance network models are better at predicting the shape of the polarization spectra. The results of this thesis work open new opportunities for clay characterization using SIP