Nätverksdynamik

**Valid from:** Spring 2017**Decided by:** Professor Thomas Johansson**Date of establishment:** 2016-10-27

**Division:** Automatic Control**Course type:** Course given jointly for second and third cycle**The course is also given at second-cycle level with course code:** FRTN30**Teaching language:** English

The course provides 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.

*Knowledge and Understanding*

For a passing grade the doctoral student must

- 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

*Competences and Skills*

For a passing grade the doctoral student must

- 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 use computer tools for simulation and analysis of networks

*Judgement and Approach*

For a passing grade the doctoral student must

- 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

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.

D. Easley & J. Kleinberg: Networks, crowds and markets, reasoning about a highly connected world. Cambridge University Press, 2010, ISBN: 978-0-521-19533-1. Supplement to lecturer's notes. R. Van Der Hofstad: Random Graphs and Complex Networks. Supplement to lecturer's notes. Tillgänglig online via http://www.win.tue.nl/~rhofstad/. D. Levin, Y. Peres, E. Wilmer: Markov chains and mixing times. American Mathematical Society, 2009, ISBN: 978-0-8218-4739-8. Supplement to lecturer's notes.

**Types of instruction:** Lectures, laboratory exercises, exercises

**Examination formats:** Written exam, written assignments.
Skriftlig examen, fyra godkända inlämningsuppgifter.**Grading scale:** Failed, pass**Examiner:**

**Assumed prior knowledge:** FRT010 Automatic Control, Basic Course

**Course coordinator:** Giacomo Como `<giacomo.como@control.lth.se>`