Hours:
16 hours (4 credits)

Room:

Aula Riunioni del Dipartimento di Ingegneria dell’Informazione, Via G. Caruso 16, Pisa - Ground Floor

To register to the course, click here

Short Abstract:

Nowadays, complex networks can be found in a wide range of scientific areas, from biology to sociology, from psychology to information science. Some of them, like the Internet, or the social networks, have even become pervasive in everyday life. Analyzing this kind of networks can reveal important features or properties, but also shed light on some peculiar behaviors which can be empirically observed in real scenarios. At the same time, the analysis and modeling of complex networks is often quite involved. Fortunately, there exist reliable mathematical tools which make it possible to build simple and effective models, and to obtain, through elegant analysis, useful and accurate results. These models can be applied to any kind of networks, and are therefore potentially useful for a number of applications (network design, studying epidemics spreading, analyzing social networks behaviors etc.). The main aim of the course is to offer an introduction on some useful mathematical tools suitable for analyzing complex networks, while giving examples on how these methods can be applied to real networks.

Course Contents in brief:

1. Mathematical preliminaries, definition of tools, metrics and measures (adjacency matrix, degree, assortativity, homophily...);
2. Models of Random Graphs, their properties (degree distribution, clustering coefficient...), their components (Giant component and Small component, distributions of their size, appearance of the Giant component...). Path lengths, “small world effect”, drawbacks of Random Graphs when applied to real networks
3. Random Graphs with general degree distribution (power law distribution, Configuration Model, Small World model, Exponential Random Graphs)
4. Models of Network Formation (Preferential Attachment, Barabasi model and its properties)
5. Basic Percolation Theory (bond and edge percolation, appearance of components, phase transition behavior)
6. Epidemics on Networks (SI, SIR, SIS and SIRS models)
7. Elements of Stochastic Geometry and its applications (Point Process Theory, Poisson Processes, Cox Processes, Gibbs Processes, Moment Generating functionals, Probability Generating Functionals, characterization of interference and outage in Poisson networks)

Schedule:

1. 06/05/26 – 9:00-13:00
2. 13/05/26 – 9:00-13:00
3. 20/05/26 – 9:00-13:00
4. 27/05/26 – 9:00-13:00