NĂ¤tverksdynamik

**Valid from:** Autumn 2013**Decided by:** FN1/Anders Gustafsson**Date of establishment:** 2014-02-07

**Division:** Automatic Control**Course type:** Third-cycle course**Teaching language:** English

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.

*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 practically apply the theory during a project
- 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
- be able to read and critically examine an article from the current literature on the subject

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.

TBD.

Lecture notes

**Types of instruction:** Lectures, seminars, exercises, project

**Examination formats:** Written assignments, seminars given by participants**Grading scale:** Failed, pass**Examiner:**

**Assumed prior knowledge:** 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, ...

**Course coordinators:**