Recent experiments suggest that the human genome (all of our DNA) is organised as a so-called fractal globule. The fractal globule is a knot--free dense polymer that easily folds and unfolds any genomic locus, for example a group of nearby genes. Proteins often need to locate specific target sites on the DNA, for instance to activate a gene. To understand how proteins move through the DNA polymer, we simulate diffusion of particles through a fractal globule. The fractal globule was generated on a cubic lattice as spheres connected by cylinders. With the structure in place, we simulate particle diffusion and measure how their mean squared displacement ($\langle R^2(t)\rangle$) grows as function of time $t$ for different particle radii. This quantity allows us to better understand how the three dimensional structure of DNA affects the protein's motion. From our simulations we found that $\langle R^2(t)/t\rangle$ is a decaying function when the particle is sufficiently large. This means that the particles diffuse slower than if they were free. Assuming that $\langle R^2(t) \rangle \propto t^\alpha$ for long times, we calculated the growth exponent $\alpha$ as a function of particle radius $r_p$. When $r_p$ is small compared to the average distance between two polymer segments $d$, we find that $\alpha \approx 1$. This means the polymer network does not affect the particle's motion. However, in the opposite limit $r_p\sim d$ we find that $\alpha<1$ which means that the polymer strongly slows down the particle's motion. This behaviour is indicative of sub-diffusive dynamics and has potentially far reaching consequences for target finding processes and biochemical reactions in the cell.

The purpose of this research is to investigate properties of diffusion processes in porous media. Porous media are modelled by random Sierpinski carpets, each carpet is constructed by mixing two different generators with the same linear size. Diffusion on porous media is studied by performing random walks on random Sierpinski carpets and is characterized by the random walk dimension $d_w$. In the first part of this work we study $d_w$ as a function of the ratio of constituents in a mixture. The simulation results show that the resulting $d_w$ can be the same as, higher or lower than $d_w$ of carpets made by a single constituent generator. In the second part, we discuss the influence of static external fields on the behavior of diffusion. The biased random walk is used to model these phenomena and we report on many simulations with different field strengths and field directions. The results show that one structural feature of Sierpinski carpets called traps can have a strong influence on the observed diffusion properties. In the third part, we investigate the effect of diffusion under the influence of external fields which change direction back and forth after a certain duration. The results show a strong dependence on the period of oscillation, the field strength and structural properties of the carpet.

The purpose of this research is to investigate properties of diffusion processes in porous media. Porous media are modelled by random Sierpinski carpets, each carpet is constructed by mixing two different generators with the same linear size. Diffusion on porous media is studied by performing random walks on random Sierpinski carpets and is characterized by the random walk dimension $d_w$. In the first part of this work we study $d_w$ as a function of the ratio of constituents in a mixture. The simulation results show that the resulting $d_w$ can be the same as, higher or lower than $d_w$ of carpets made by a single constituent generator. In the second part, we discuss the influence of static external fields on the behavior of diffusion. The biased random walk is used to model these phenomena and we report on many simulations with different field strengths and field directions. The results show that one structural feature of Sierpinski carpets called traps can have a strong influence on the observed diffusion properties. In the third part, we investigate the effect of diffusion under the influence of external fields which change direction back and forth after a certain duration. The results show a strong dependence on the period of oscillation, the field strength and structural properties of the carpet.

Networks that self-organize in response to information are one of the most central studies in complex systems theory. A new time series analysis tool for studying self-organizing systems is developed and demonstrated. This method is applied to interacting complex swarms to explore the connection between information transport and group size, providing evidence for Dunbar's numbers having a foundation in network dynamics. A complex network model of information spread is developed. This network infodemic model uses reinforcement learning to simulate connection and opinion adaptation resulting from interaction between units. The model is applied to study polarized populations and echo chamber formation, exploring strategies for network resilience and weakening. The model is straightforward to extend to multilayer networks and networks generated from real world data. By unifying explanation and prediction, the network infodemic model offers a timely step toward understanding global collective behavior.

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No description available.

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.

Passeio aleatório é uma classe de modelos matemáticos que têm por objetivo descrever processos estocásticos cujo resultado observável é dado por uma soma de variáveis aleatórias. O termo foi cunhado em 1905 pelo estatístico inglês Karl Pearson, estando na época interessado na modelagem da migração de insetos, e hoje possui uma ampla gama de aplicações, indo desde a biologia, passando pela física e química, e chegando na economia. Tendo sido estudado por inúmeros cientistas, muitas variações surgiram, chegando aos passeios aleatórios correlacionados, processos estocásticos não-Markovianos nos quais as variáveis aleatórias que se somam, chamadas de passos, possuem dependências umas com as outras, com correlações de caudas longas. Em 2004, surge na literatura o passeio aleatório do elefante, um passeio aleatório correlacionado com um mecanismo microscópico de memória de longo alcance muito bem definido e com soluções analíticas. Além desses dois fatos, também despertou o interesse da comunidade científica por exibir superdifusão. Muitas variações desse modelo foram propostas e vários resultados foram obtidos nos anos que se seguiram. A presente dissertação contem uma compilação dos principais modelos e resultados da área, tentando ser um texto introdutório ao assunto, focando sempre no que diz respeito à difusão. No caso unidimensional, propomos uma generalização desse tipo de passeio aleatório, o qual envolve decisões probabilísticas com respeito a passos lembrados do passado. Já no caso multi-muldimensional, apresentamos o conceito de acoplamento de memória e o modelo de Vaca e Boi, introduzidos pelo autor deste trabalho em 2018, como uma maneira de incluir interações entre elefantes. Também obtivemos um limite do contínuo para esse último processo, permitindo calcular os regimes de difusão para o Boi e construir um diagrama de fases para o mesmo. Esses últimos pontos constituem as principais contribuições do presente trabalho. / Random walk is a class of mathematical models which has the objective of describing a stochastic process whose observable result is given by a sum of aleatory variables. The term was coined in 1905 by the english statistician Karl Pearson while he was interested in the insects migration modeling, but today it has a myriad of applications, from biology to stock markets, passing through physics and chemestry. It has been studied by an uncountable number of scientists and a lot of variations have appeared, including those called correlated random walks, which are stochastic non-Markovian process in which those random variables that are summed, called steps, depends one of each other with fat tails correlations. In 2004, the elephant random walk appeared in the literature. It is a correlated random walk with a microscopic well defined memory mechanism and that has analitical solutions. Besides these facts, it also arouse the interest of scientific community because it exhibits superdifusion behaviour. In the one-dimensional case, we propose a generalization of this kind of random walk, which involves probabilistic decisions with respect to remembered steps given in the past. In the multi-dimensional case, we present the concept of memory coupling and the Cow and Ox model, which were introduced by the author of this work in 2018 as a manner of including interactions among elephants. We have also obtained a continuum limit of this process, allowing us to calculate the Ox diffusion regimes and to build its phase diagram. These last points constitute the main contributions of the present work.

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Considere um meio desordenado formado por $N$ pontos cujas coordenadas são geradas aleatoriamente com probabilidade uniforme ao longo das arestas unitárias de um hipercubo de $d$ dimensões. Um caminhante, partindo de um ponto qualquer desse meio, se desloca seguindo a regra determinista de dirigir-se sempre ao ponto mais próximo que não tenha sido visitado nos últimos $\\mu$ passos. Esta dinâmica de movimentação, denominada caminhada determinista do turista, leva a trajetórias formadas por uma parte inicial transiente de $t$ pontos, e uma parte final cíclica de $p$ pontos. A exploração do meio se limita aos $t+p$ pontos percorridos na trajetória. O sucesso da exploração depende do valor da memória $\\mu$ do viajante. Para valores pequenos de $\\mu$ a exploração é altamente localizada e o sistema não é satisfatoriamente explorado. Já para $\\mu$ da ordem de $N$, aparecem ciclos longos, permitindo a exploração global do meio. O objetivo deste estudo é determinar o valor de memória $\\mu_1$ para o qual ocorre uma transição abrupta no comportamento exploratório do turista em meios unidimensionais. Procuramos também entender a distribuição da posição final do turista após atingir um estado estacionário que é atingido quando o turista fica aprisionado nos ciclos. Os resultados obtidos por simulações numéricas e por um tratamento analítico mostram que $\\mu_1 = \\log_2 N$. O estudo também mostrou a existência de uma região de transição com largura $\\varepsilon = e/ \\ln 2$ constante, caracterizando uma transição aguda de fase no comportamento exploratório do turista em uma dimensão. A análise do estado estacionário da caminhada em função da memória mostrou que, para $\\mu$ distante de $\\mu_1$, a dinâmica de exploração ocorre como um processo difusivo tradicional (distribuição gaussiana). Já para $\\mu$ próximo de $\\mu_1$ (região de transição), essa dinâmica segue um processo superdifusivo não-linear, caracterizado por distribuições $q$-gaussianas e distribuições $\\alpha$-estáveis de Lévy. Neste processo, o parâmetro $q$ funciona como parâmetro de ordem da transição. / Consider a disordered medium formed by $N$ point whose coordinates are randomly generated with uniform probability along the unitary edges of a $d$-dimensional hypercube. A walker, starting to walk from any point of that medium, moves following the deterministic rule of always going to the nearest point that has not been visited in the last $\\mu$ steps. This dynamic of moving, called deterministic tourist walk, leads to trajectories formed by a initial transient part of $t$ points and a final cycle of $p$ points. The exploration of the medium is limited to the $t+p$ points covered. The success of the exploration depends on the traveler\'s memory value $\\mu$. For small values of $\\mu$, the exploration is highly localized and the whole system remains unexplored. For values of $\\mu$ of the order of $N$, however, long cycles appear, allowing global exploration of the medium. The objective of this study is to determine the memory value $\\mu_1$ for which a sharp transition in the exploratory behavior of the tourist in one-dimensional media occurs. We also want to understand the distribution of the final position of the tourist after reaches a steady state in exploring the medium. That steady state is reached when the tourist is trapped in cycles. The results achieved by numerical simulations and analytical treatment has shown that $\\mu_1 = \\log_2 N$. The study has also shown the existence of a transition region, with a constant width of $\\varepsilon = e/ \\ln 2$, characterizing a phase transition in the exploratory behavior of the tourist in one dimension. The analysis of the walk steady state as a function of the memory has shown that for $\\mu$ far from $\\mu_1$, the exploratory dynamic follows a traditional diffusion process (with gaussian distribution). In the other hand, for $\\mu$ near $\\mu_1$ (transition region), the dynamic follows a non-linear superdiffusion process, characterized by $q$-gaussian distributions and Lèvy $\\alpha$-stable distributions. In this process, the parameter $q$ plays the role of a transition order parameter.