This thesis utilizes statistical physics concepts and mathematical modeling to study complex systems. I investigate the emergent complexities in two systems: (i) the stock volume volatility in the United States stock market system; (ii) the robustness of networks in an interdependent lattice network system. In Part I, I analyze the United States stock market data to identify how several financial factors significantly affect scaling properties of volume volatility time intervals. I study the daily trading volume volatility time intervals between two successive volume volatilities above a certain threshold q, and find a range of power law distributions. I also study the relations between the form of these distribution functions and several financial factors: stock lifetimes, market capitalization, volume, and trading value. I find that volume volatility time intervals are short-term correlated. I also find that the daily volume volatility shows a stronger long-term correlation for sequences of longer lifetimes. In Part II, I apply percolation theory to interacting complex networks. The dependency links between the two square lattice networks have a typical length r lattice units. For two nodes connecting by a dependency link, one node fails once the node on which it depends in the other network fails. I show that rich phase transition phenomena exist when the length of the dependency links r changes. The results suggest that percolation for small r is a second-order transition, and for larger r is a first-order transition. The study suggests that interdependent infrastructures embedded in two-dimensional space become most vulnerable when the interdependent distance is in the intermediate range, which is much smaller than the size of the system.

Le schéma générique de percolation 1-liaison→2-TO mis au point sur site pour la compréhension de base des spectres de vibration (Raman et infrarouge - IR) des alliages semiconducteurs aléatoires usuels, i.e., s’inscrivant dans la structure zincblende, qui a récemment supplanté le modèle standard 1-liaison→1-TO dit MREI [modified-random-element-isodisplacement, (Chang, et al., 1971)] mis au point dans les années soixante, est ici confronté au modèle à clusters (Verleur, et al., 1966), consacré par l’usage pour la discus-sion des spectres de vibration de ces alliages zincblende particuliers, présumés non aléatoires, qui montrent évidemment plus d’un mode phonon par liaison dans leur spectres de vibration. En fait, nous montrons que les spectres de réflectivité IR pionniers de l’alliage GaAs1-xPx représenta-tif de cette dernière classe de comportement, i.e. ceux-là mêmes qui ont motivé le développement du modèle à clusters dans les années soixante, trouvent une explication naturelle dans le cadre du modèle de percolation sur la base d’une substitution As ↔ P parfaitement aléatoire, sans aucun paramètre ajustable. L’alliage GaAs1-xPx et ses semblables se trouvent ainsi réhabilités en tant que systèmes aléatoires vis-à-vis de leurs propriétés vibrationnelles. Du même coup, la classification admise des spectres de vibration des alliages semiconducteurs à structure zincblende, basée sur les modèles MREI-Clusters, en quatre sous-types distincts, se trouve totalement unifiée dans le cadre de notre modèle de percolation. Dans un second temps nous explorons dans quelle mesure une extension du schéma de percolation de la structure zincblende vers la structure diamant, peut aider à comprendre la nature mystérieuse du comportement Raman de l’alliage aléatoire représentatif Si1-xGex. Il s’avère qu’une simulation satisfaisante des spectres Raman de Si1-xGex d’ores et déjà existants dans la littérature peut être accomplie, sans autre para-mètre ajustable que l’efficacité Raman de la liaison mixte Si-Ge, dans le cadre d’une version générique 1-liaison→N-mode du schéma de percolation, où N indique une sensibilité à l’environnement local des liaisons Ge-Ge (N=1, insensibilité), Si-Si (N=2, sensibilité à l’environnement premiers-voisins) et Si-Ge (N=3, sen-sibilité à l’environnement seconds-voisins). Des différences notables entre les schémas de percolation de GaAs1-xPx et de Si1-xGex sont attribuées à la différence de nature de la relaxation du réseau dans les cristaux à structure zincblende et diamant (inversion de l’ordre des branches au sein du triplet Si-Ge) et à la dispersion spectaculaire des modes Ge-Ge (antiparallélisme de la branche unique Ge-Ge) et Si-Si (inversion de l’ordre des branches au sein du doublet Si-Si), qui vient s’ajouter à l’effet de contrainte locale, habituellement seul pris en compte pour les alliages zincblende déjà examinés. L’assignation des branches phonons individuelles au sein de chaque schéma de percolation, i.e. GaAs1-xPx ou Si1-xGex, est réalisé par voie ab initio en se focalisant sur les modes de vibration des liaisons en stretching pur le long de motifs d’impuretés prototypes choisis quasi-linéaires de manière à rester dans l’esprit de l’approximation de la chaîne linéaire (ACL) sur laquelle est basé le modèle phénoménologique de percolation / The 1-bond → 2-TO percolation generic scheme proposed for the basic understanding of the vibra-tion spectra (Raman and Infrared –IR) of usual random semiconductor alloys with zincblende structure, which has recently challenged the standard MREI (modified-random-element-isodisplacement) model with the 1-bond→1-TO behavior, (Chang et Mitra, 1971) developped in the sixties, is here confronted to the cluster model (Verleur et Barker, 1966) which has been accepted through use for the study of the vibration spectra of these particular zincblende alloys, which obviously exhibit more than one phonon mode per bond in their vibration spectra. In fact, the IR reflectivity spectra of GaAs1-xPx, the representative alloy of the last class, i.e. those very ones which motivated the developpement of the cluster model in the sixties, find a natural explanation in the scope of the percolation model on the basis of a perfect random substitution As ↔ P, with no adjusta-ble parameter. With this, GaAs1-xPx and its like are rehabilitated as random alloys in principle, and further, the percolation paradigm generalizes to all types and subtypes of the traditional four-type classification of phonon mode behavior of semiconductor alloys, based on the MREI and cluster models. In a second stage, we investigate to which extent an extension of the percolation scheme to the di-amond structure may help to understand the mysterious nature of the Raman behavior of the representative Si1-xGex alloy. We realize that a good simulation of the Si1-xGex Raman spectra already existing in the literature can be obtained with no adjustable parameter but the Raman efficiency of the Si1-xGex bond within a generic 1-bond→N-TO version of the percolation model, where N indicates a sensitivity to the local environment of the Ge-Ge (N=1 non sensitive), Si-Si (N=2, sensitive to the first neighbor environment) and Si-Ge (N=3, sensitive to the second neighbor environment) bonds. Some differences between the GaAs1-xPx and Si1-xGex percolation patterns are attributed to the different natures of the lattice relaxation in the zincblende and di-amond structures (inversion of the order of like branches in the Si-Ge triplet) and to the spectacular disper-sion of the Ge-Ge like modes (anti parallelism of the unique Ge-Ge branch) and Si-Si (inversion of the order of like branches in the Si-Si doublet), which add to the local constraint effect, the only one usually taken into account in the zincblende alloys examined so far. The assignment of the individual phonons branches in each percolation scheme, i.e. GaAs1-xPx or Si1-xGex, is achieved via home-made ab initio phonon calculations with a focusing on bond-stretching along prototype impurity motives designed as quasi linear so as to be remain in with the spirit of the linear chain approximation on which the percolation model is based

In this dissertation we consider several extremal and probabilistic problems in bootstrap percolation on various families of graphs, including grids, hypercubes and trees. Bootstrap percolation is one of the simplest cellular automata. The most widely studied model is the so-called r-neighbour bootstrap percolation, in which we consider the spread of infection on a graph G according to the following deterministic rule: infected vertices of G remain infected forever and in successive rounds healthy vertices with at least r already infected neighbours become infected. Percolation is said to occur if eventually every vertex is infected. In Chapter 1 we consider a particular extremal problem in 2-neighbour bootstrap percolation on the n \times n square grid. We show that the maximum time an infection process started from an initially infected set of size n can take to infect the entire vertex set is equal to the integer nearest to (5n^2-2n)/8. In Chapter 2 we relax the condition on the size of the initially infected sets and show that the maximum time for sets of arbitrary size is 13n^2/18+O(n). In Chapter 3 we consider a similar problem, namely the maximum percolation time for 2-neighbour bootstrap percolation on the hypercube. We give an exact answer to this question showing that this time is \lfloor n^2/3 \rfloor. In Chapter 4 we consider the following probabilistic problem in bootstrap percolation: let T be an infinite tree with branching number \br(T) = b. Initially, infect every vertex of T independently with probability p > 0. Given r, define the critical probability, p_c(T,r), to be the value of p at which percolation becomes likely to occur. Answering a problem posed by Balogh, Peres and Pete, we show that if b \geq r then the value of b itself does not yield any non-trivial lower bound on p_c(T,r). In other words, for any \varepsilon > 0 there exists a tree T with branching number \br(T) = b and critical probability p_c(T,r) < \varepsilon. However, in Chapter 5 we prove that this is false if we limit ourselves to the well-studied family of Galton--Watson trees. We show that for every r \geq 2 there exists a constant c_r>0 such that if T is a Galton--Watson tree with branching number \br(T) = b \geq r then \[ p_c(T,r) > \frac{c_r}{b} e^{-\frac{b}{r-1}}. \] We also show that this bound is sharp up to a factor of O(b) by describing an explicit family of Galton--Watson trees with critical probability bounded from above by C_r e^{-\frac{b}{r-1}} for some constant C_r>0.

A coated conductor is generally fabricated by depositing a high Tc superconducting layer onto a flexible metallic substrate, using intermediate buffer layers to prevent chemical interaction. In order for the superconductor to be capable of carrying a high current density, its grains must have good crystallographic alignment in order to avoid the presence of high angle grain boundaries. This can be ensured by transferring the texture from the substrate through epitaxial film growth. The main substrate considered in this thesis is a Ni-Fe alloy. When cold-rolled, NiFe develops a preferential orientation and upon annealing at an elevated temperature, undergoes primary recrystallisation to form grains with the cube texture {100}<001>. There crystallisation process and the texture of the tapes has been examined and various buffer layers have been fabricated. As silver does not react adversely with high temperature superconductors, it has been deposited onto Pd-buffered NiFe by DC sputtering and very sharp cube texture is obtained. Ceramic buffer layers, CeO2 and YSZ, have been deposited by RF sputtering, though an undesirable (111) oriented component accompanies the cube textured material. Also a technique has been developed to produce a suitably oriented native oxide of NiFe by a simple oxidation technique. Preliminary attempts to deposit YBCO films onto these buffer layers have shown that the quality of the metallic buffers is degraded by rapid inter-diffusion at elevated temperatures, but that cube textured material can be deposited on the oxide buffer layers. The percolative nature of current flow in such coated conductors has been considered through the development of a grain network model. As the texture of the superconducting layer is directly influenced by the underlying layers, measurements from the substrate and buffer layers are applied in order to model the orientations of the grains in a superconducting overlayer. The model calculates the critical current of coated conductors as a function of parameters such as length, width, grain size and texture, as well as examining factors such as cracks and highly misoriented grains.

Complex networks appear in almost every aspect of our daily life and are widely studied in the fields of physics, mathematics, finance, biology and computer science. This work utilizes percolation theory in statistical physics to explore the percolation properties of complex networks and develops a reinforcement scheme on improving network resilience. This dissertation covers two major parts of my Ph.D. research on complex networks: i) probe—in the context of both traditional percolation and k-core percolation—the resilience of complex networks with tunable degree distributions or directed dependency links under random, localized or targeted attacks; ii) develop and propose a reinforcement scheme to eradicate catastrophic collapses that occur very often in interdependent networks. We first use generating function and probabilistic methods to obtain analytical solutions to percolation properties of interest, such as the giant component size and the critical occupation probability. We study uncorrelated random networks with Poisson, bi-Poisson, power-law, and Kronecker-delta degree distributions and construct those networks which are based on the configuration model. The computer simulation results show remarkable agreement with theoretical predictions. We discover an increase of network robustness as the degree distribution broadens and a decrease of network robustness as directed dependency links come into play under random attacks. We also find that targeted attacks exert the biggest damage to the structure of both single and interdependent networks in k-core percolation. To strengthen the resilience of interdependent networks, we develop and propose a reinforcement strategy and obtain the critical amount of reinforced nodes analytically for interdependent Erdős-Rényi networks and numerically for scale-free and for random regular networks. Our mechanism leads to improvement of network stability of the West U.S. power grid. This dissertation provides us with a deeper understanding of the effects of structural features on network stability and fresher insights into designing resilient interdependent infrastructure networks.

A multiscale continuum model for fragmentation in ductile metals was developed, motivated by structure-property relationships obtained from lower length scale and numerical simulations. Fragmentation occurs during high strain rate deformation as the result of widespread internal damage in the form of void or crack nucleation, growth, and coalescence. The connection between internal damage structures and fragmentation was determined through Molecular Dynamics (MD) simulations of high rate deformation in copper, iron, and iron-carbon alloys. The fragmentation metric of interest in this study is the fragment size, which is represented in MD simulations by the fragment length scale, or the solid volume per surface area ratio. Three deformation modes of varying stress triaxialities, plane strain tension, equibiaxial expansion, and isotropic expansion, provide a range of damage growth behavior allowing the fragment length scale to be correlated to damage structures under different conditions. Modified Embedded Atom Method (MEAM) potentials for the materials enable the representation of damage (and newly created free surfaces) under the extreme conditions. Continuum, nonhomogeneous percolation simulations establish a criterion for fragmentation based on internal damage structure. The continuum percolation simulations are motivated by void size and shape information taken from experimental fracture surfaces of an aluminum 7085 alloy. The combination of the percolation based fragmentation criterion and MD motivated fragmentation model yields a framework for the multiscale modeling of fragmentation.

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Fractures are ubiquitous in the subsurface, and they provide dominant pathways for fluid flow in low permeability formations. Therefore, fractures usually play an essential role in many engineering fields, such as hydrology, waste disposal, geothermal reservoir and petroleum reservoir exploitation. Since fractures are invisible and have variable sizes from micrometers to kilometers, there is limited knowledge of their structure. We aim to deepen the understanding of fracture networks in the subsurface from their topological structures, hydraulic connectivity and characteristics at different scales. We adopt the discrete fracture network method and develop an efficient C++ code, HatchFrac, to make in-depth investigations possible. We start from generating stochastic fracture networks by constraining fracture geometries with different stochastic distributions. We apply percolation theory to investigate the global connectivity of fracture networks. We find that commonly adopted percolation parameters are unsuitable for the characterization of the percolation state of complex fracture networks. We implement the concept of global efficiency to quantify the impact of fracture geometries on the connectivity of fracture networks. Furthermore, we constrain the fracture networks with geological data and geomechanics principles. We investigate the correlation of fracture intensities with different dimensionality and find that it is not feasible to obtain correct 3D intensity parameters from 1D or 2D samples. We utilize a deep-learning technique and propose a pixel-based detection algorithm to automatically interpret fractures from raw outcrop images. Interpreted fracture maps provide abundant resources to investigate fracture intensities, lengths,orientations, and generations. For large scale faults, we develop a method to generate fault segments from a rough fault trace on a seismic map. Accurate fault geometries have significant impacts on damage zones and fault-related flow problems. For small scale fractures, we consider the impact of fracture sealing on the percolation state of orthogonal fracture networks. We emphasize the importance of non-critically stressed and partially sealed fractures, which are usually neglected because usually they are nonconductive. However, with significant stress perturbations, those noncritically stressed and partially sealed fractures can also contribute to the production by enlarging the stimulated reservoir volume.

The study of three-dimensional percolation paths through materials is important in its contribution to understanding defect sensitive properties of materials. This work shows the importance of grain boundary character in modeling defect sensitive boundaries. Also presented are trends of percolation of sensitized grain boundaries in 304 stainless steel (304SS). Of particular interest is how open paths form in a three-dimensional model created through serial sectioning. Evidence is presented that triple or quadruple points that contain typically two boundaries with special character that intersect the percolation path break up the path. Some grains with no known special qualities (i.e., CSL) have observable special behavior.