English

If sufficient demand

This course will provide an introduction to and some analysis of the main mathematical models used to describe large networks and dynamical processes that evolve on networks. Motivation and applications will be drawn from social, economic, natural, and infrastructure networks, as well as networked decision systems such as sensor networks.

Basic graph theory: connectivity, degree distributions, trees, adjacency matrices, spectrum.
Random graphs: Erdos-Renyi, configuration model, preferential attachment, small-world, branching process approximations
Flows and games on graphs: max-flow min-cut, optimal transport, Wardrop equilibria, evolutionary dynamics.
Random walks on graphs: invariant distributions, hitting times, mixing times.
Dynamical systems on graphs: distributed averaging, interacting particle systems, epidemics, opinion dynamics. Mean-field and branching process approximations.

know the basic principles of graph theory and apply them to model real-world networks
have insight in the basic differences between different models of random graphs
be familiar with the properties of random walks on graphs
be able to analyze simple dynamical systems over networks
understand emerging phenomena in large-scale networks
be able to give an overview of modern directions in network science

be able to analyze properties of (random) graphs both quantitatively and qualitatively
be able to handle basic analytical computations for random walks
be able to analyze simple dynamical systems over networks and to relate their behavior to the network structure
be able to practically apply the theory during a project
be able to use computer tools for simulation and analysis of networks

be able to understand relations and limitations when simple models are used to describe complex networks
be able to evaluate dominating emerging phenomena in network dynamics
be able to read and critically examine an article from the current literature on the subject

Lectures

Seminars

Exercises

Project

Written assignments

Seminars given by participants

Failed, pass

The course is designed for an audience of doctoral students with some mathematical background (in particular, probability and linear algebra at an intermediate level), possibly from the programs in Automatic Control, Mathematics, Mathematical Statistics, Physics, Computer Science, Economics, ...

TBD.

Lecture notes

FRT095F

 -02-07

FN1/Anders Gustafsson