Unit name | Complex Networks |
---|---|

Unit code | MATH36201 |

Credit points | 20 |

Level of study | H/6 |

Teaching block(s) |
Teaching Block 1 (weeks 1 - 12) |

Unit director | Dr. Ayalvadi Ganesh |

Open unit status | Not open |

Pre-requisites |
MATH11300 Probability 1 (or the first half of MATH10013 Probability and Statistics), MATH11005 Linear Algebra and Geometry and MATH20008 Probability 2 is strongly recommended. |

Co-requisites |
None |

School/department | School of Mathematics |

Faculty | Faculty of Science |

**Unit Aims**

Networks are very widely used in mathematical modelling. They can describe physical systems such as transportation or telecommunication networks, or interactions between agents such as in social networks of humans or other biological organisms. It is of interest to study both the structure of the network, and dynamical processes occurring on a network.

Motivated by such questions, this unit will introduce a number of different random processes to model information spread, consensus formation, and random walks on networks. We will also study random network models and some of their properties.

The course will emphasise both proofs and applications.

**Unit Description**

This unit will teach ways of modelling and working with large networks such as the Internet and social networks. The topics covered will be:

- Probability background: Continuous time Markov chains and Poisson processes
- Spread of information and epidemics on networks
- Consensus formation on networks
- Random walks on networks and introduction to spectral graph theory
- Random graph models and properties

**Relation to Other Units**

The unit applies basic probabilistic models studied in Probability 2, specifically Markov chains and martingales, to the study of random processes on networks.

Graph theory would also have been introduced in Combinatorics. This unit does not have significant overlap with Combinatorics but takes a complementary approach to studying graphs using probability.

The course provides an interesting applied context for deepening the student's understanding of probabilistic techniques learnt in other courses such as Probability 2 and Martingale Theory and Applications.

Learning Objectives

- Learn to model a variety of stochastic processes on graphs, including random walks and the spread of information and epidemics
- Learn to analyse these processes to obtain bounds and approximations for quantities of interest
- Learn about the relationship of graph spectra to various properties of the graph

Lectures and problem sheets, from which work will be set and marked, with outline solutions handed out a fortnight later.

10% Coursework

90% Examination

Raw scores on the examinations will be determined according to the marking scheme written on the examination paper. The marking scheme, indicating the maximum score per question, is a guide to the relative weighting of the questions. Raw scores are moderated as described in the Undergraduate Handbook.

**Recommended**

- Moez Draief and Laurent Massoulie,
*Epidemics and Rumours in Complex Networks*, LMS Lecture Note Series 369, Cambridge University Press, 2009 - Richard Durrett,
*Random Graph Dynamics*, Cambridge University Press, 2007 - Devavrat Shah, G
*ossip Algorithms*, NOW Publishers Inc., 2009