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.

### Aims and Objectives

#### Module Aims

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.

#### Learning Outcomes

##### Learning Outcomes

Having successfully completed this module you will be able to:

• Define basic network properties, compute them in theoretical and practical situations, and explain their significance in network modelling
• Understand and develop your capacity to plan and manage your own learning
• Demonstrate an ability to write clearly and accurately a mathematical essay in a given theme in network theory, relating and coherently integrating different literary sources, and showing some degree of originality in the exposition, treatment, examples or applications
• Explain the relation between some network structural and spectral properties, and their significance for real-world networks
• Express stochastic processes mathematically as a dynamic equation for the probability of stochastic outcomes.
• Explain the generic relationship between stochastic processes and networks.
• Extract information about the probability distribution of stochastic processes, in particular of stochastic processes on networks.

### 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, tutorials, guided reading and private study. The lectures will be based on selected material from the reading list. Lectures will give an overview of the topic and introduce the main references and students are expected to demonstrate in-depth independent learning through private study. The module is organised in three blocks of four weeks each.

TypeHours
Independent Study102
Teaching48
Total study time150

#### Resources & Reading list

M.E.J.Newman (2010). Networks: An Introduction.

A.-L.Barabasi (2016). Network Science.

### Assessment

#### Summative

MethodPercentage contribution
In-class Test  (40 minutes) 70%
Report or Essay 30%

#### Referral

MethodPercentage contribution
Final Exam 100%

#### Repeat Information

Repeat type: Internal & External

We use cookies to ensure that we give you the best experience on our website. If you continue without changing your settings, we will assume that you are happy to receive cookies on the University of Southampton website.

×