# The scientific project:

modeling, simulation, prediction, control

The key distinctive and original feature of this project is a strong focus on the mathematics of complex networks. In all the problems that we consider, networks either appear as the primary research objects or as natural and useful modeling tools. Networks can be used to model infrastructural systems, social and economic relationships, protein interaction schemes, underground fracture architectures, or chemical reaction systems. In the mathematics of complex networks, deterministic and random formalism often coexist and typically pose significant big data statistical problems in their tuning and reliable use. The presence of time and space multi-scales, as well as the simultaneous presence of heterogeneous geometrical objects, are other features of these models. Their investigation calls for a sophisticated combination of different mathematical tools. The analysis of models composed of a hierarchical architecture of sub-models and sub-structures is one of the scientific aims of this project.

The following are the main research topics of the project:

- T1. Resilient control for network systems
- T2. Nested mathematical models in biomedicine
- T3. Numerical methods for models with high geometric complexity
- T4. Approximation and statistical inference in random reaction (and interaction) networks

#### T1. Resilient control for network systems

It is well known that the malfunctioning of big infrastructural networks can cause major social effects limiting the access to essential services like mobility and energy, influencing the outcome of electoral polls and possibly destabilizing large economic systems. A central feature of such networked systems is the role that interconnections play in propagating and amplifying perturbations even if small or localized (systemic risk). The term "resilience" refers to the capability of a system to limit the propagation and the effect of such disturbances, thus maintaining an acceptable functionality. The ambitious goal of this project is to develop notions of dynamic resilience, through which one can easily predict the effect of perturbations on a large-scale network and optimize its behavior.

Contact person: Giacomo Como

#### T2. Nested mathematical models in biomedicine

In biology and medicine, the macroscopic behavior of a living system is intrinsically related to phenomena that take place at a microscopic level. Mathematical models must, therefore, incorporate the dynamics of events that occur at different spatial and temporal scales, making them intrinsically multi-scale (multi-level) mathematical problems. For example, to study phenomena that operate at the tissue or cellular scale, one must take into account and model processes that take place at the sub-cellular level. This project will explore, in various contexts, the interaction of systems operating at different scales. The idea is, therefore, to use mathematical models as a sort of virtual microscope, focusing on the cellular or sub-cellular levels to extract the key features of their role at a macro-scale in a formally precise averaged form.

Contact person: Davide Ambrosi

#### T3. Numerical methods for models with high geometric complexity

A source of serious complexity in numerical modeling and simulation is the presence of dimensionally inhomogeneous geometric structures, which play a fundamental role in many physical phenomena. Examples are: reinforcing fibers in homogeneous materials, networks of (possibly evolving) fractures in the subsoil, capillaries or fibrous structures in biological tissues. Their numerical treatment with standard approaches is often unfeasible or prohibitively expensive; for instance, the process of mesh generation to conform to small geometrical structures may yield a vast number of elements. The project aims to explore recently proposed alternatives to classical approaches. They have been successfully applied to some of these problems and are based on constrained optimization techniques that provide flexibility and computational efficiency. Furthermore, in this context, we will develop uncertainty quantification methods to perform stochastic analyses when there is substantial uncertainty on the geometry or the material properties. To achieve these goals, it will be crucial to investigate geometrical and algebraic theoretical issues.

Contact person: Stefano Berrone

#### T4. Approximation and statistical inference in random reaction (and interaction) networks

Several phenomena in biology, finance, telecommunications, economy and the social sciences can be modeled with interconnected Markov chains. The complexity of such phenomena is caused by a large number of interacting agents, which may render the corresponding mathematical models hardly applicable in practice. The project aims to a) find effective approximated models, and b) calibrate stochastic models against experimental data by extracting information from the noise and not only from the mean signal.

Contact person: Enrico Bibbona