In the final quarter of the XX century the classical reductionist approach that had been driving the development of physics was questioned. Instead, it was proposed that systems were arranged in hierarchies so that the upper level had to convey to the rules of the lower level, but at the same time it could also exhibit its own laws that could not be inferred from the ones of its fundamental constituents. This observation led to the creation of a new field known as complex systems. This novel view was, however, not restricted to purely physical systems. It was soon noticed that very different systems covering a huge array of fields, from ecology to sociology or economics, could also be analyzed as complex systems. Furthermore, it allowed physicists to contribute with their knowledge and methods in the development of research in those areas.
In this thesis we tackle problems covering three areas of complex systems: networks, which are one of the main mathematical tools used to study complex systems; epidemic spreading, which is one of the fields in which the application of a complex systems perspective has been more successful; and the study of collective behavior, which has attracted a lot of attention since data from human behavior in huge amounts has been made available thanks to social networks. In fact, data is also the main driver of our discussion of the other two areas. In particular, we use novel sources of data to challenge some of the classical assumptions that have been made in the study of networks as well as in the development of models of epidemic spreading.
In the case of networks, the problem of null models is addressed using tools coming from statistical physics. We show that anomalies in networks can be just a consequence of model oversimplification. Then, we extend the framework to generate contact networks for the spreading of diseases in populations in which both the contact structure and the age distribution of the population are important.
Next, we follow the historical development of mathematical epidemiology and revisit the assumptions that were made when there was no data about the real behavior of this kind of systems. We show that one of the most important quantities used in this kind of studies, the basic reproduction number, is not properly defined for real systems. Similarly, we extend the theoretical framework of epidemic spreading on directed networks to multilayer systems. Furthermore, we show that the challenge of incorporating data to models is not only restricted to the problem of obtaining it, but that it is also really important to be aware of its characteristics to do it properly.
Lastly, we conclude the thesis studying two examples of collective behavior using data extracted from online systems. We do so using techniques that were originally developed for other purposes, such as earthquake prediction. Yet, we demonstrate that they can also be used to study this new type of systems. Furthermore, we show that, despite their unique characteristics, they possess properties similar to the ones that have been observed in the offline world. This not only means that modern societies are intertwined with the online world, but it also signals that if we aim to understand socio-technical systems a holistic approach, as the one proposed by complex systems, is indispensable.