Venue: Seminar Room "-1" – Via Sommarive, 14 Povo - Trento

Hour: 4.00 p.m.

- Giulia Bertagnolli -
*PhD**in Mathematics*

### Abstract:

A group of individuals sharing posts on climate exchange on Twitter, following other users and re-tweeting other's opinion can be represented mathematically using a graph. A graph G consists of a set of vertices V (the individuals) and a set of edges E⊆V × V (interactions among individuals, e.g. re-tweet). In the most general setting a graph G is described by V and a by a function w : V × V → I ⊆ ℝ. This led to the generalisation of (finite) graphs to graphons, graph-functions, W : [0, 1]^{2} → [−1, 1] with a continuum of vertices in [0, 1]. Apart from being interesting for the study of limiting properties of large networks, they enable the extension of Mean Field Games (MFGs) to the case of non-uniform spatial distribution of agents. In the classical MFG framework the actions of the individual players, among a very large number of players, do not affect each other's strategy directly, but only through the population's mean. During this seminar, after becoming familiar with the graphon terminology, we will go through the Graphon MFG (GMFG) framework proposed by Caines and Huang in 2018, where the network connectivity creates local mean fields, and the steps leading to the existence theorem for solutions of the GMFG equations.

*The seminar corresponds to part of the Ph.D. exam of the course “Mean Field Games and Optimal Transport” *

Contact person:

Fabio Bagagiolo