Abstract: This dissertation articulates concepts from different disciplines and incorporates a comprehensive study of complex systems. Many dynamical processes on complex structures of interactions exhibit forms of emergent behaviors, often unspecified in the equations of motion. In consequence, this thesis can be divided into two parts: First, we introduce many basic concepts of Network Science, essential to understand the main findings of our research. Second, we present all of the results of nonlinear dynamics on top of synthetic and real graphs, providing a comparative analysis between the behaviors of our simulated configurations and the phenomena observed in electronic, neuronal and social systems. In short, once both constituents have been implemented and optimized, the chief goal is to investigate and analyze thoroughly the models of nonlinear dynamics in such systems, incorporating aspects such as analytical predictions, the appearance of collective complex behaviors. SYSTEMS OF INTEREST - Networks: The interest in the study of complex networked systems has been steadily increasing over the past few decades, especially in Physics as a results of the prominent advantages provided by the standard techniques of the statistical mechanics. In our case, we will focus on discrete systems that can be modeled as static complex networks. As an emblematic example of structural complexity let us comment on the scale-free topology. As its name indicates, these networks do not posses a characteristic scale (their degree distribution is a power-law), allowing certain structures to appear in all the scales (notion that connects with both fractals and the scaling observed in phase transitions). These emergent topological properties, can be interpret as a collective behavior in terms of the so-called dynamical models (generating algorithms that involve growth), which is precisely one of the universal fingerprints of complexity. As an illustration of their versatility, in the scientific literature there are numerous applications and significant results in various disciplines; ranging from physics (e.g. cosmology, quantum mechanics, materials, etc.) and engineering (e.g. Internet, communication and power networks) to biology (e.g. proteins, genetics, neuroscience, epidemics, etc.) and sociology (e.g. real and online social networks, small world effect and authors’ network, among others). - Synchronization: Among all of the emergent behaviors that have been found in Complex Systems as yet, synchronization stands out for its ubiquitous nature. This phenomenon was originally observed by Christiaan Huygens, in the mid-17th century, from a set of coupled pendulum clocks, and it is still one of the most enigmatic outcomes detected in all kinds of coupled oscillating setting. Remarkable real-world examples of synchrony include flashing of fireflies, electric activity in the brain (epileptic seizures), activity patterns of cardiac cells (pacemakers), optical communications, arrays of electric and electronic circuits, and a broad family of dynamical systems (which includes, of course, chaotic oscillators). Regarding this last example, that is, systems consisting of units operating in chaotic regime, there has been made an enormous effort in developing a vast amount of research over the past three decades. Many of these developments aimed to establish a general framework that could account for and classify the numerous kinds of synchronized states reported in the literature, whereas others to introduce novel mathematical indicators of synchrony. At the network level a lot of works have also been devoted to studying synchronization phenomena, particularly on diverse types of deterministic oscillators, with outstanding rigorous findings that have spread to other disciplines as a consequence of their practical implications. Although a lot of research has been done, mostly to elucidate the intricate mechanisms underlaying the diverse types of synchronization reported thus far, many questions remain open to date, especially for networked systems. In our particular case, we will examine thoroughly two types of synchronization, complete and generalized, in a elementary configuration that allows that two separate units communicate –indirectly– through a third one (in which the channels of interactions are instantaneous and bidirectional). This setting has never been considered by other authors to study such forms of synchrony, but there are many reasons perform such analysis given its the possible implications to both experimental and natural systems like electronic circuits, Lasers, underwater acoustic sensors and, the Brain. Our sound results include the determination of the existence of Relay Synchonization for the unconnected units, which is characterised by one single positive Lyapunov exponent. Similarly, we were able to prove that Generalized synchronization takes place right at the coupling where the identical units exhibit complete synchonization. - Neural dynamics: In living systems of neurons there are many kinds of cell architectures, and that they all exhibit a broad range of behaviors (diverse regimes of activity, firing mechanisms, adaptation, spikes, and so on). Over the years our models of individual neurons have evolved into a high-level of sophistication as an attempt to capture such diversity, and nowadays it is possible to perform extensive and expensive simulations on large populations of these independent units. Regarding the interactions among real neurons, it is indeed an essential and elusive aspect to reproduce realistically, yet networks provide a useful and simple framework to this end. A great deal of research has been devoted precisely to study emergent properties of ensembles of neurons that interact according to both simple and complex topologies, a topic that continues to be an attractive interdisciplinary subject for physicists, mathematicians and neuroscientists. Most of these works capitalize on numerical approaches that have demonstrated to be quite versatile; one can generate various underlying structures of interactions and include many realistic ingredients like chemical synapses, ion currents, compartments, stochastic currents, noise and adaptative parameters. However, numerical models also have many limitations and, at the network level, analytical results are scarce, particularly for sparsely connected systems (which resemble the actual structure of the brain). To illustrate this point, that is the lack of rigorous results on networks, let us consider some of the most common constraints imposed on the system of study. As far as we know, the vast majority of the analytical approaches already published for spiking models of neurons require some sort of diffusion approximation, so that the membrane potential can be described approximately as a random process. For this attribute to be true, the density of links of the network has to be high, which, by definition, is not the case for sparse network. A second mathematical technique is the so-called “spike- train statistics”, which is a framework developed to analyze and fit spike patterns of population of neurons by means of a characterizing probability function. Even though this attempt seems promising, especially to study activity patterns of living neurons, it is a proposal that is far from fully developed (there is no general strategy to derive such probability function from data, and the precise mathematical form of it is required –as well as a procedure to fit accurately the many parameters that are needed). Finally, concerning the popular family of neural dynamics known as integrate-and- fire, numerous studies devoted to numerical explorations were performed over the past four decades, whereas just a handful were published on developing analytics (mostly based on mean-field approximations). These limited theoretical developments also hinge on similar assumptions as the ones mentioned above, namely densely connected networks or stochastic approximations (useless for sparse graphs). In contrast, in the present thesis we will examine large ensembles of excitable elements on heterogeneous topologies, providing both numerical findings and novel analytical estimations to decompose the activity into connectivity groups and to predict average quantities that are essential to describe global activity. - Socio-technical environment: The proliferation of social networking tools –and the massive amounts of data associated to them– has evidenced that modeling social phenomena demands a complex, dynamic perspective. Physical approaches to social modeling are contributing to this transition from the traditional paradigm (scarce data and/or purely analytical models) towards a data-driven new discipline. This shift is also changing the way in which we can analyze social contagion and its most interesting consequence: the emergence of information cascades in the Information and Communication Technologies (ICT) environment. Theoretical approaches, like epidemic and rumor dynamics, reduce these events to physically plausible mechanisms. These idealizations deliver analytically tractable models, but they attain only a qualitative resemblance to empirical results, for instance regarding cascade size distributions. The vast majority of models to this end –including the thresh- old model, overviewed in this dissertation– are based on a dynamical process that determines individuals’ activity (transmission of information), and this activity is propagated according to certain rules usually based on the idea of social reinforcement, i.e. the more active neighbors an individual has, the larger his probability to become also active, and thus to contribute to the transmission of information. Yet, the challenge of having mechanistic models that include more essential factors, like the self-induced (intrinsic, spontaneous) propensity of individuals to transmit information, still remains open –though some contributions emerge in this fast-growing field. Furthermore, the availability of massive amounts of microblogging data logs, like Twitter, places scholars in the position to scrutinize the patterns of real activity and model them. These patterns indicate that avalanche phenomena are not isolated events. Instead, users engaged in a certain topic repeatedly participate, affecting each other and giving rise to an heterogeneous collection of cascades emerging over time, which can not be modeled independently from each other. As a consequence of these observations, in this thesis we propose a dynamical threshold model that is able to display a broad variety of behaviors and reproduce the activity patterns detected in online platforms, extending our comprehension of social interactions and our ability to predict the occurrence of system-wide events by means of a useful cascade conditions that allows to determine in advance macroscopic events.