The University of Southampton
Courses

# MATH6140 Structure and Dynamics of Networks

## Module Overview

Networks are ubiquitous in the modern world: from the biological networks that regulate cell behaviour, to technological networks such as the Internet and social networks such as Facebook. Typically real-world networks are large, complex, and exhibit both random and regular properties, making them both challenging and interesting to model. This course is an introduction to the structure and dynamics of networks, with emphasis on real-world applications. The first part of the course is an introduction to the structure of networks: from basic network properties and terminology, to network eigenvalues and eigenvectors. The second part of the course is on dynamics of networks: network generative models such classical random graph models, and basic models of growing networks (small-worlds, Albert-Barabasi); and dynamics on networks: random walks and synchronization phenomena. In the third part, we will focus on modelling the stochastic dynamics that characterize many biological regulatory networks, including a discussion of the chemical master equation, its approximations and methods of simulation of stochastic dynamics

### Aims and Objectives

#### Module Aims

This course is an introduction to the structure and dynamics of networks, with emphasis on real-world applications.

#### Learning Outcomes

##### Learning Outcomes

Having successfully completed this module you will be able to:

• Understand and develop your capacity to plan and manage your own learning
• Demonstrate an ability to write a mathematical essay clearly and accurately within a given scope
• Relate and coherently integrate different literary sources
• Show some degree of originality in the exposition, treatment, examples or applications
• Effectively manage time when working towards a deadline
• Choose a topic and propose a tentative content for a short essay within a given theme in network theory
• Demonstrate in-depth independent learning on a topic within a proposed theme in network theory

### Syllabus

Part I: Network structure and eigenvalues • Network terminology • Network eigenvalues and eigenvectors, and their relation to structural network properties Part II: Dynamics of and on networks • Generative network models: random graphs, small worlds, Barabasi-Albert model • Examples of dynamical processes on networks: random walks, models of coupled oscillators Part III: Stochastic dynamics on networks • Overview of general stochastic differential equations, including the chemical master equation • Biochemical reaction networks

### Learning and Teaching

#### Teaching and learning methods

Lectures and tutorials, organised in three blocks of four weeks each (2 weeks for lectures followed by 2 weeks for tutorials), guided literature reading, private study

TypeHours
Teaching30
Independent Study120
Total study time150

A.L.Barabasi. Network Science.

M.E.J.Newman. Networks: an introduction.

### Assessment

#### Assessment Strategy

Late coursework will receive a deduction of 10% of the marks for each university working day after the deadline.

#### Summative

MethodPercentage contribution
Coursework assignment(s) 100%

#### Referral

MethodPercentage contribution
Coursework assignment(s) 100%

#### Repeat Information

Repeat type: Internal & External