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Topics for Master and PhD thesis
Dynamics over large-scale networks.

Socio-economic, financial, biological, or infrastructural (electric grids, transportation) networks represent some of the most challenging and exciting multi-disciplinary scientific areas of these decades. They are of large scale, with a complex dynamical behavior emerging from the interconnection of relatively simple units. The presence of nonlinearities and stochastic effects, as well the complexity of the topology of interconnections make their analysis a formidable task and is catalyzing the energies of a large number of scientists with expertise ranging from economics, theoretical engineering, to mathematics and physics.

The typical way to describe such complex systems is that of a population of units (agents) interacting according to a communication graph. Each unit is equipped with a state variable which may be discrete or continuous. Depending on the application, the state may be interpreted as an opinion, an economic value, an index of functionality, a measurement or an estimation of an environmental quantity, or rather a physical quantity like a position or a potential. Each unit modifies its state in time as the effect of some endogenous dynamics as well of the interaction with the neighbor units in the communication graph. Mathematically, this leads to a dynamical system evolving in the phase space consisting of all the state vectors (assembling the state of the various units). Depending on the assumptions on the local dynamics and interactions, this dynamical systems may be deterministic, or, rather, a stochastic process. Typically, such local dynamics are simple, but the complexity emerges from the global network coupling effects.

A survey paper on these topics is:

Castellano, Fortunato, Loreto: Statistical physics of social dynamics

Specific arguments for research thesis: Some of these topics constitute current joint research with Daron Acemoglu (MIT), Giacomo Como (Lund University), Jean-Charles Delvenne (Louvain), Asu Ozdaglar (MIT).