Demyelination is the process where the insulating layer of axons known as myelin is damaged. This affects the propagation of action potentials along axons which can have deteriorating consequences on the motor activity of an organism. Thus it is important to understand the biophysical effects of demyelination to improve the diagnostics of its diseases. We trained a Convolutional Neural Network (CNN) on Coherent anti-Stokes Raman scattering (CARS) microscope images of mice spinal cord inflicted with the demyelinating disease Experimental Autoimmune Encephalomyelitis (EAE). Our CNN was able to classify the images reliably based on clinical scores assigned to the mice. We then synthesized our own images of the spinal cord regions using a 2D Biased Random Walk. These images are simplified versions of the original CARS images and show homogenously myelinated axons, unlike the heterogeneous nerve fibres found in real spinal cords. The images were fed into the trained CNN as an attempt to develop a clinical connection to the biophysical effects of demyelination. We found that the trained CNN was indeed able to capture structural features related to demyelination which can allow us to constrain demyelination models such that they include the simulated parameters of the synthesized images.

Graphs are a great tool for representing complex physical and social systems, where the interactions among many units, from tens of animal species in a food-web, to millions of users in a social network, give rise to emergent, complex system behaviours. In the field of network science this representation, which is usually called a complex network, can be complicated at will to better represent the real system under study. For instance, interactions may be directed or may differ in their strength or cost, leading to directed weighted networks, but they may also depend on time, like in temporal networks, or nodes (i.e. the units of the system) may interact in different ways, in which case edge-coloured multi-graphs and multi-layer networks represent better the system. Besides this rich repertoire of network structures, we cannot forgot that edges represent interactions and that this interactions are not static, but are, instead, purposely established to reach some function of the system, as for instance, routing people and goods through a transportation network or cognition, through the exchange of neuro-physiological signals in the brain. Building on the foundations of spectral graph theory, of non-linear dimensionality reduction and diffusion maps, and of the recently introduced diffusion distance [Phys. Rev. Lett. 118, 168301 (2017)] we use the simple yet powerful tool of continuous-time Markov chains on networks to model their process-driven geometry and characterise their functional shape. The main results are: (i) the generalisation of the diffusion geometry framework to different types of interconnected systems (from edge-coloured multigraphs to multi-layer networks) and of random walk dynamics [Phys. Rev. E 103, 042301 (2021)] and (ii) the introduction of new descriptors based on the diffusion geometry to quantify and describe the micro- (through the network depth [J. Complex Netw. 8, 4 (2020)]), meso- (functional rich-club) and macro-scale (using statistics of the pairwise distances between the network's nodes [Comm. Phys. 4, 125 (2021)]) of complex networks.

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Quantum random walks are a generalization of classical Markovian random walks to a quantum mechanical or quantum computing setting. Quantum walks have promising applications but are complicated by quantum decoherence. We prove that the long-time limiting behavior of the class of quantum operations which are the convex combination of norm one operators is governed by the eigenvectors with norm one eigenvalues which are shared by the operators. This class includes all operations formed by a coherent operation with positive probability of orthogonal measurement at each step. We also prove that any operation that has range contained in a low enough dimension subspace of the space of density operators has limiting behavior isomorphic to an associated Markov chain. A particular class of such operations are coherent operations followed by an orthogonal measurement. Applications of the convergence theorems to quantum walks are given. / Mathematics

In this master thesis we will show how random walks are used in Wilson's algorithm to generate spanning trees of graphs, and how they can be used to calculate the number of spanning trees in a graph. We will also explore the connection between electrical networks and random walks, and how this connection can be exploited to prove Pólya's theorem (normally proved with probability and combinatorics) using electrical arguments.

Rotational-translational decoupling, a phenomenon commonly observed in supercooled liquids, has been a topic of great interest. Despite its prevalence, the underlying cause of this phenomenon, often attributed to dynamic heterogeneity, has not been conclusively elucidated. This thesis investigates and evaluates how dynamic heterogeneity may lead to this decoupling using simultaneous single-molecule rotational and translational measurements. In the experimental study, single molecule fluorescence imaging experiments are performed on the ideal probe N,N’-dipentyl-3,4,9,10-perylenedicarboximide in high molecular weight polystyrene near its glass transition temperature. A novel trajectory linking method based on hierarchical clustering is developed to facilitate single molecule tracking even in imaging data where specific molecules cannot be observed visually for a substantial number of frames. This linking algorithm then allows molecules to be localized over full movies, such that rotational and translational measurements can be compared over comparable timespans. The investigation of translational dynamics using such long trajectories, which was not previously achieved, reveals that both rotational-translational decoupling and translational enhancement persist on the single molecule level, supporting the hypothesis that temporally heterogeneous dynamics experienced by the probe molecules is a contributing factor in observed rotational-translational breakdown in both ensemble and single molecule studies. A tendency towards dynamical convergence between subgroups with fast and slow dynamics is observed, demonstrating temporal heterogeneity at the single molecule level. In comparison to rotational dynamics, translational dynamics was discovered to have a longer lifetime. Other key observations facilitated by the linked trajectory analysis include that apparent diffusion coefficient of probe molecules decreases with longer observation time, a finding inconsistent with normal diffusive behavior. To investigate the origin of this anomalous slowing in single molecule studies existing alongside the observed overall enhancement in translational motion, temporally heterogeneous models with multiple types of correlation were studied via simulations. The results emphasize the critical role that bias in translational and rotational measurements can play when investigating and observing dynamic heterogeneity, as nearly all models including dynamic heterogeneity show increasing diffusion coefficient with increasing number of dynamic environments explored. Strikingly, translational enhancement is evident in single molecule translational simulations even when slow dynamics are reinforced via positive correlation in the models. A comparison of the diffusion coefficient evolution between simulations and experiments reveals that the sub-diffusive continuous time random walk model is the most plausible candidate to account for the set of observations seen in experiment.

In this dissertation, I extend the analysis of human dynamics to human movements in online activities. My work starts with a discussion of the human information foraging process based on three large collections of empirical search click-through logs collected in different time periods. With the analogy of viewing the click-through on search engine result pages as a random walk, a variety of quantities like the distributions of step length and waiting time as well as mean-squared displacements, correlations and entropies are discussed. Notable differences between the different logs reveal an increased efficiency of the search engines, which is found to be related to the vanishing of the heavy-tailed characteristics of step lengths in newer logs as well as the switch from superdiffusion to normal diffusion in the diffusive processes of the random walks. In the language of foraging, the newer logs indicate that online searches overwhelmingly yield local searches, whereas for the older logs the foraging processes are a combination of local searches and relocation phases that are power-law distributed. The investigation highlights the presence of intermittent search processes in online searches, where phases of local explorations are separated by power-law distributed relocation jumps. In the second part of this dissertation I focus on an in-depth analysis of online gambling behaviors. For this analysis the collected empirical gambling logs reveal the wide existence of heavy-tailed statistics in various quantities in different online gambling games. For example, when players are allowed to choose arbitrary bet values, the bet values present log-normal distributions, meanwhile if they are restricted to use items as wagers, the distribution becomes truncated power laws. Under the analogy of viewing the net change of income of each player as a random walk, the mean-squared displacement and first-passage time distribution of these net income random walks both exhibit anomalous diffusion. In particular, in an online lottery game the mean-squared displacement presents a crossover from a superdiffusive to a normal diffusive regime, which is reproduced using simulations and explained analytically. This investigation also reveals the scaling characteristics and probability reweighting in risk attitude of online gamblers, which may help to interpret behaviors in economic systems. This work was supported by the US National Science Foundation through grants DMR-1205309 and DMR-1606814. / Ph. D. / Humans are complex, meanwhile understanding the complex human behaviors is of crucial importance in solving many social problems. In recent years, socio physicists have made substantial progress in human dynamics research. In this dissertation, I extend this type of analysis to human movements in online activities. My work starts with a discussion of the human information foraging process. This investigation is based on empirical search logs and an analogy of viewing the click-through on search engine result pages as a random walk. With an increased efficiency of the search engines, the heavy-tailed characteristics of step lengths disappear, and the diffusive processes of the random walkers switch from superdiffusion to normal diffusion. In the language of foraging, the newer logs indicate that online searches overwhelmingly yield local searches, whereas for the older logs the foraging processes are a combination of local searches and relocation phases that are power-law distributed. The investigation highlights the presence of intermittent search processes in online searches, where phases of local explorations are separated by power-law distributed relocation jumps. In the second part of this dissertation I focus on an in-depth analysis of online gambling behaviors, where the collected empirical gambling logs reveal the wide existence of heavy-tailed statistics in various quantities. Using an analogy of viewing the net change of income of each player as a random walk, the mean-squared displacement and first-passage time distribution of these net income random walks exhibit anomalous diffusion. This investigation also reveals the scaling characteristics and probability reweighting in risk attitude of online gamblers, which may help to interpret behaviors in economic systems. This work was supported by the US National Science Foundation through grants DMR-1205309 and DMR-1606814.

Dans cette thèse on s'intéresse au temps local d'auto-intersections de marches aléatoires. Cette quantité est définie comme la norme-p à la puissance p du temps local de la marche. Elle regarde dans quelle mesure la trajectoire de la marche aléatoire s'intersecte. Le temps local d'auto-intersections est lié à différents modèles physiques comme les modèles de polymères ou les problèmes d'écoulements de flux en milieux stratifiés mais aussi au modèle mathématiques des marches aléatoires en paysages aléatoires. Nous nous sommes pour notre part intéressés en particulier aux grandes déviations du temps local d'auto-intersections, c'est à dire que nous regardons la probabilité que la quantité d'intersections de la marche aléatoire soit plus grande que sa moyenne. Cette question qui a été très étudiée au cours des années 2000 fait apparaitre trois cas distincts, le cas sous-critique, le cas critique et le cas sur-critique. Nous améliorons la connaissance sur cette question au travers de deux résultats complets et d'un résultat partiel. D'abord nous prouvons un principe de grandes déviations dans les cas critique et sur-critique des marches alpha-stables, puis nous améliorons les échelles de déviations au cas sous-critique tout entier de la marche simple, enfin nous sommes en train d'étendre ce dernier résultat aux marches alpha-stables. Par ailleurs les trois preuves sont basées sur l'utilisation d'une version due à Eisenbaum d'un théorème d'isomorphisme de Dynkin. Cette méthode d'abord introduite par Castell dans le cas critique est donc ici étendue aux autres cas. Nous avons donc réussi à unifier les différentes méthodes de preuves au travers ce théorème d'isomorphisme. / In this thesis we are interested in the self-intersection local times of random walks. This quantity is defined as the p-norm to the power of p of the local times of the random walk. It measures how much the trajectory of the random walk intersects itself. The self-intersection local times is connected with various physical models as polymer models or problems of anomalous dispersion in layered random flows, but it is also linked with the mathematical model of random walks in random sceneries. More precisely, we are interested in the large deviations of the self-intersection local times, i.e. we work on the probability for the intersections to be larger than expected. This question that has been studied a lot during the 2000's is divided in three cases, the subcritical one, the critical one and the super critical one. We improve the knowledge about this question by two complete results and a partial one. First, we have proved a large deviation principle in the critical and super critical cases of alpha-stable random walks, then we have improved the deviations' scales to the entire subcritical case of simple random walk, finally we are extending this last result to the alpha-stable random walks. The three proofs are based on a version due to Eisenbaum of a Dynkin isomorphism theorem. This method which has been first introduced by Castell in the critical case, is extended here to the others cases. Thus, we have succeeded to unify the methods of proof by this isomorphism theorem.

The financial market is a stochastic and complex system that is challenging to model. It is crucial for investors to be able to model the probability of possible outcomes of financial investments and financing decisions in order to produce fruitful and productive investments. This study investigates how Monte Carlo simulations of random walks can be used to model the probability of future stock returns and how the simulations can be improved in order to provide better accuracy. The implemented method uses a mathematical model called Geometric Brownian Motion (GBM) in order to simulate stock prices. Ten Swedish large-cap stocks were used as a data set for the simulations, which in turn were conducted in time periods of 1 month, 3 months, 6 months, 9 months and 12 months. The two main parameters which determine the outcome of the simulations are the mean return of a stock and the standard deviation of historical returns. When these parameters were calculated without weights the method proved to be of no statistical significance. The method improved and thereby proved to be statistically significant for predictions for a 1 month time period when the parameters instead were weighted. By varying the assumptions regarding price distribution with respect to the size of the current time period and using other weights, the method could possibly prove to be more accurate than what this study suggests. Monte Carlo simulations seem to have the potential to become a powerful tool that can expand our abilities to predict and model stock prices. / Den finansiella marknaden är ett stokastiskt och komplext system som är svårt att modellera. Det är angeläget för investerare att kunna modellera sannolikheten för möjliga utfall av finansiella investeringar och beslut för att kunna producera fruktfulla och produktiva investeringar. Den här studien undersöker hur Monte Carlo-simuleringar av så kallade random walks kan användas för att modellera sannolikheten för framtida aktieavkastningar, och hur simuleringarna kan förbättras för att ge bättre precision. Den implementerade metoden använder den matematiska modellen Geometric Brownian Motion (GBM) för att simulera aktiepriser. Tio svenska large-cap aktier valdes ut som data för simuleringarna, som sedan gjordes för tidsperioderna 1 månad, 3 månader, 6 månader, 9 månader och 12 månader. Huvudparametrarna som bestämmer utfallet av simuleringarna är medelvärdet av avkastningarna för en aktie samt standardavvikelsen av de historiska avkastningarna. När dessa parametrar beräknades utan viktning gav metoden ingen statistisk signifikans. Metoden förbättrades och gav då statistisk signifikans på en 1 månadsperiod när parametrarna istället var viktade. Metoden skulle kunna visa sig ha högre precision än vad den här studien föreslår. Det är möjligt att till exempel variera antagandena angående prisernas fördelning med avseende på storleken av den nuvarande tidsperioden, och genom att använda andra vikter. Monte Carlo-simuleringar har därför potentialen att utvecklas till ett kraftfullt verktyg som kan öka vår förmåga att modellera och förutse aktiekurser.

Thesis (MSc)--Stellenbosch University, 2014. / ENGLISH ABSTRACT: We study random walks on nite graphs. The reader is introduced to general Markov chains before we move on more specifically to random walks on graphs. A random walk on a graph is just a Markov chain that is time-reversible. The main parameters we study are the hitting time, commute time and cover time. We nd novel formulas for the cover time of the subdivided star graph and broom graph before looking at the trees with extremal cover times. Lastly we look at a connection between random walks on graphs and electrical networks, where the hitting time between two vertices of a graph is expressed in terms of a weighted sum of e ective resistances. This expression in turn proves useful when we study the cover cost, a parameter related to the cover time. / AFRIKAANSE OPSOMMING: Ons bestudeer toevallige wandelings op eindige gra eke in hierdie tesis. Eers word algemene Markov kettings beskou voordat ons meer spesi ek aanbeweeg na toevallige wandelings op gra eke. 'n Toevallige wandeling is net 'n Markov ketting wat tyd herleibaar is. Die hoof paramaters wat ons bestudeer is die treftyd, pendeltyd en dektyd. Ons vind oorspronklike formules vir die dektyd van die verdeelde stergra ek sowel as die besemgra ek en kyk daarna na die twee bome met uiterste dektye. Laastens kyk ons na 'n verband tussen toevallige wandelings op gra eke en elektriese netwerke, waar die treftyd tussen twee punte op 'n gra ek uitgedruk word in terme van 'n geweegde som van e ektiewe weerstande. Hierdie uitdrukking is op sy beurt weer nuttig wanneer ons die dekkoste bestudeer, waar die dekkoste 'n paramater is wat verwant is aan die dektyd.