Stochastic binary-state dynamics on complex networks: a theoretical analysis

Peralta, Antonio F. (supervisor: Raúl Toral)
PhD Thesis (2020)

Binary-state models on complex networks are built as a simplified theoretical construction to analyze the stochastic dynamics of a system made of interacting individuals. The individuals that compose these systems are assumed to hold a binary state, whose interpretation depends on the context of the model, that evolves in time following a determined set of rules. Typical examples that can be studied using this tool include: epidemic spreading (the two options being infected or susceptible to the illness), language competition (speaking language A or language B), opinion dynamics (supporting option A or option B), financial markets (selling or buying state of a broker), among many others. Usually, these models display a phase transition where a change of the macroscopic behavior is expected, for example: from healthy to endemic phases in epidemic spreading or from coexistence to consensus in opinion dynamics. The characterization of these transitions and an understanding of how and when they emerge is one of the main goals of the models, and it can be carried out using some tools of statistical mechanics.

The aim of this thesis is to deepen on the theoretical methods to study binary-state models and to explore the role of each of the more relevant ingredients that define their dynamical rules, specifically: the effect of network structure in the interactions, non-linear state copying mechanisms and non-Markovia (memory) effects. We analyze different approximate techniques, as well as their accuracy and suitability to elucidate their effects on the macroscopic dynamics. Among these we consider in detail sophisticated compartmental approach theories, where the specific details of the interactions are aggregated into a few number of variables, thus greatly improving the simplicity of the description.

The thesis is presented as a compendium of publications, where each chapter of the central part of the thesis is a different published contribution, or in process of being published. This comes preceded by an introduction chapter, where we explain the field of study, a chapter of methodology, where we review the methods used and, finally, a chapter with the conclusions. We divide the contents in three parts, where each part contains several related publications depending on the ingredient of the models that we are considering with particular attention. As listed above these are: effect of network structure, non-linear copying mechanisms and non-Makrovian dynamics.

Network structure, which determines the interaction relations between individuals, is one of the crucial features in the definition of the models. It influences most of the stationary and dynamical results such as: the existence, the nature and the location of the phase transitions, the size of the fluctuations and the time scale of the evolution of the global state of the system. The evaluation of these effects is a non-trivial task as they strongly depend on the rules of the model. We investigate how to take into account network structure in a full stochastic description of binary-state models, at different levels of approximation using mean field theories. The methods developed can be applied to any model and, only for some simple models of interest, we are able to obtain an explicit analytical solution that enlarges our understanding of the dynamics.

Among all the mechanisms that one can postulate and that may produce a change of state of the individuals, we focus on two prototypical ones: (i) imitating or copying the state of one or more other individuals, and (ii) changing state randomly independently of the others. This may correspond to a standard opinion dynamics scenario. The simplest mechanism that one can study is blind imitation, which corresponds to pairwise state copying between interacting individuals. We explore different generalizations of this rule by including a non-linear copying mechanisms, which can be understood as a group instead of pairwise interaction. The impact of this generalization is analyzed, taking into account also the effect of network structure. The results are far from trivial, exhibiting the appearance of induced continuous and discontinuous coexistence-consensus transitions whose properties are well characterized using theoretical and numerical methods, including some of the mean field approaches studied in the first part of the thesis.

Memory (non-Markovian) effects are also to be taken into account as an important feature of the models. The idea is to consider that the rules that drive the dynamics depend, not only on the present set of binary states of the population but also, at some extend, of previous states. This may represent an aging related property of the individuals, where the longer an individual spends holding the same state the harder it is to change state. We develop a theoretical basis to include this factor in the mathematical description of the dynamics, which enables us to characterize the macroscopic behavior of these systems. Some of these models can be mapped to an equivalent ``effective'' memory-less (Markovian) model, while others require a more demanding analysis, and we determine under which conditions the reduction is possible. Examples of non-Markovian models that can not be reduced are those displaying burstiness, power-law interevent-time distributions, or those whose dynamics get trapped in a frozen state, both are analyzed in detail.

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