Advanced Methods for Complex Systems I

This interdisciplinary course aims at introducing rigorous tools from statistical physics, information theory and probability theory to investigate real-world complex systems arising in different fields of research. First, some key aspects of complexity encountered in physical, biological, social, economic and technological systems will be reviewed. Then, emphasis will be put on the construction of theoretical models based on the concept of constrained randomness, i.e. the maximisation of the entropy subject to suitable constraints. This will lead to the introduction of maximum-entropy models that serve as mathematical benchmarks for the properties of highly heterogeneous complex systems. Special cases of interest include statistical ensembles of complex networks and of multivariate time-series with given properties. Comparisons between model outcomes and empirical properties will be presented systematically. Full mathematical derivations of the models, as well as methods of statistical inference, model selection and computer codes for parameter estimation on empirical data will be provided. The course will include a combination of recent and ongoing research in the NETWORKS unit at IMT Lucca, thereby offering directions for possible PhD projects in this area.