Network science is one of the most challenging and exciting fields of interdisciplinary research. Applications are ubiquitous and include infrastructure networks (e.g., power grid, transportation networks, water distribution, supply chains), socio-economic networks, financial networks, and biological networks. These networks are characterized by their large scale and potentially complex dynamical behaviors emerging from the interconnection of many relatively simple units.

Mathematically, such network dynamics are described by a large set of units, each characterized by a certain state (typically, a scalar, a vector, or a discrete variable). States are updated as a result of repeated interactions over a graph. Interactions are typically simple and can be described either as deterministic, or as stochastic processes, depending on the application. E.g., in opinion dynamics, each unit is meant to represent an individual, whose state describes her/his opinion on a certain topic of interest; interaction occurs over a social network where nodes represent individuals and links between two nodes represent the possibility that the corresponding individuals meet and exchange their opinions. In urban transportation networks, units are meant to represent roads in a city, their states are the vehicle densities, and the dynamics comes from conservation of mass and the routing of drivers choosing which road to take next when reaching a junction.

There is plenty of room here for great creative research. Here are some specific themes for master theses. Some of these may well be explored in higher depth and become a starting point for a PhD theses.

Multi-commodity dynamical flows

Mean-field approximations

Control of cascades

Opinion fluctuations and the role of minorities in social networks