Complex Systems and Networks

Director: Tiziano Squartini

Curriculum Overview

Many of the challenges in modern society require the understanding and management of the complexity of physical, biological, social, economic, financial and technological networks. Besides the traditional difficulties inherent in the study of the components of these systems (such as atoms, cells, individuals, organizations, devices), over the last decades an additional (and often dominant) level of complexity has emerged, which derives from the interactions among the components of a system. In a world that is increasingly interconnected at the physical, social, cultural, economic and digital levels, the bodies of knowledge developed by individual scientific disciplines are becoming less and less exhaustive, and the need for innovative interdisciplinary approaches emerges forcefully.

The Theory of Complex Systems and the Science of Networks are modern approaches to the study of complex systems characterized by a large number of heterogeneous interacting components that are interconnected in irregular architectures - i.e., structures that are quite different from those traditionally considered in the natural and social sciences. Indeed, while the interactions among atoms in simple materials can be represented as regular and symmetric lattices, and those among social actors or economic agents as homogeneous structures, real-world networks of interaction among the constituents of cells, organisms, ecosystems, societies, economies and infrastructures turn out to be extremely heterogeneous. Examples of recurrent structures in empirical networks are: the coexistence of elements (vertices) displaying such diverse numbers of connections that the notion of “average number of connections per node” becomes meaningless (scale-free property), the tendency of vertices with “neighbours in common” to be also connected among themselves (clustering or triadic closure), a larger cohesion within certain sets of vertices (community structure) and the abundance of specific substructures (motifs). Complex systems also exhibit collective properties that emerge from the interactions among their consituent elements and that cannot be traced back uniquely to the intrinsic properties of the latter.

Besides the need to characterise the complex structure of large-scale real-world systems, understanding the consequences of structural complexity for the dynamics of processes that typically take place on those systems has become more and more important. For instance, recent financial, economic and health crises have shown how the highly irregular and inhomogeneous structure of real networks of interaction (among banks, firms or people) deeply complicates the management (and even more so the prediction) of stress and disease propagation in modern economies and societies. Indeed, the phenomenology of these processes crucially depends on which vertices are hit first, which and how many vertices are directly connected to them, and so on - along intertwined chains of interaction. Finally, in many contexts (e.g. in ecology and economics) a strong interplay is observed between the structure of networks and the dynamics of processes taking place on it: not only network structure impacts the dynamical process, but also (and conversely) the dynamical process impacts network structure.

The PhD track in Complex Systems and Networks offers a multidisciplinary scientific background aimed at the empirical analysis, the mathematical modelling, the theoretical understanding and the development of novel methodologies for the study of complex systems, as well as their application to problems of societal relevance. The program, among the few of its kind at the international level, places theoretical research in networks and complex systems science as its core distinctive component, emphasising methodolgical innovation (such as the introduction of novel quantitative methods of analysis).

The teaching program consists in doctoral courses that cover both a wide spectrum of theoretical knowledge (graph theory, random matrices, stochastic processes, statistical physics, complex networks, information theory, dynamics on networks, machine learning, optimization) and a broad range of possible applications (to financial, economic, social, biological, neural, ecological, energetic, infrastructural systems). The theoretical methods introduced in the courses include techniques of pattern detection in empirical systems, time series analysis, network inference from partial information, physical models of complex systems and networks, noise filtering in networks and time series. The applications include problems related to financial regulation, economic resilience, sustainability, ecological stability, (mis)information diffusion, health. Besides the institutional courses, the program offers several seminars by international researchers and experts, visiting research and training periods abroad (possibly also as co-tutelle), and constant supervision from the PhD advisor(s) and the professors contributing to the track, as well as from their international collaborators.

Input and Output Profiles

Candidate PhD students who wish to carry out research oriented primarily towards theoretical modelling and methodological innovation should preferably have a background in physics, mathematics, computer science, statistics, engineering or a related field, while those who have more applied interests (to economics, biology, social science, sustainability, etc.) should preferably have a strong quatitative background in the respective field. The PhD track trains both towards an academic career (in university departments or research centres, primarily in statistical physics, applied mathematics and information theory) and towards the public sector (e.g. governmental institutions, statistical offices) and the private environment (data scientists, quants, analysts).

For more information regarding the activities and the research personnel linked to the PhD track, please visit https://networks.imtlucca.it/.

Research Units contributing to the track

Networks