Abstract: Dario Madeo (2015)

A complex system is any system featuring a large number of interacting components exhibiting hierarchical, spatially distributed and self-organizing structures (agents, processes, etc.) whose aggregate activity is driven by nonlinear mechanisms. Moreover, the dynamics of a complex system is not derivable from the summation of the activity of individual components. Although almost all processes in nature are highly cross linked, in many systems we can distinguish a set of fundamental building blocks, which interact nonlinearly to form compound structures or functions with an identity that requires more explanatory devices than those used to explain the building blocks. Examples of these systems are gene networks driving developmental processes, immune networks that preserve the identity of organisms, social insect colonies, neural networks in the brain that produce intelligence and consciousness, ecological networks, human behavior, social networks (not only Facebook!) comprised of transportation, energy and telecommunication networks, as well as economies.

Since their introduction, complex systems have been extensively studied from physical and mathematical point of views. The mathematical foundation of this discipline can be traced back to the work by H. Poincare [1] and to the pioneering studies in meteorology and ecology by Lorenz [2] and May [3]. Afterwards, the research on complexity followed two main directions, focused on system dynamics and structure. From the dynamical point of view, the discovery of complex systems led to the introduction of mathematical methods for describing unpredictable behavior and deterministic chaos [4,5] emerging from nonlinear dynamical systems. The analysis of chaotic dynamics has been also tackled by suitable techniques developed in the field of nonlinear time series analysis [6,7], which has been extensively applied to many research fields, including the analysis of physiological signals, such as human electrocardiograms and electroencephalograms. From the structural point of view, networks have been recognized to cover a fundamental role in the organization of complex systems. Network theory and graph theory have been used to describe many natural phenomena and the effects produced by the innovation introduced recently in the field of information and communication technologies, which changed our everyday life [8]. The coupling of nonlinear dynamics and networks allowed a better understanding of many unexplained phenomena, such as diffusion of epidemics, social and collective behaviors, synchronization, traffic, cell biology and brain networks [9]. More in general, complex spatial systems are those described by many variables, with high levels of interdependence between elements, governed by nonlinear processes and having significant spatial structure. Analogously to the networked case, spatially distributed systems may present complex structures produced by self-organizing mechanisms (see for example Turing instability [10]).

Although the literature on complexity science has reached a well established development, some additional issues are still uncovered. In particular, complex system theory is mainly focused on descriptive models and on their qualitative analysis. In the authorâ€™s opinion, coupling the existing methodologies with modeling, identification and control techniques could improve significantly the scientific knowledge of these systems, although these approaches are sensitive to the presence of huge numbers of variables and parameters, as well as nonlinearity of the models, nonstationarity of time series and spatiality.

The contribution of the present thesis is an attempt to cover three main topics belonging to the field of distributed complex systems jointly with modeling, identification and control. First of all, a decision based model developed within the theory of evolutionary games is introduced for studying dynamical systems composed by rational agents organized on a network of connections and driving their behavior to desired equilibria. Secondly, the application of identification techniques to gene regulatory networks represents a significant step to obtain a deeper knowledge on the distributed system, allowing the diagnosis of anomalies within a population of living cells, such as illness or harmful genetic modifications. Finally, the nonlinear time series analysis of the brain electrical activity is used for improving the knowledge on the intrinsic cerebral complexity, showing correlations between hypnotizability and cognitive tasks, such as relaxation and pain perception.

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Modeling and Identification of Networked and Distributed Complex Systems