The University of Southampton
Courses

# MATH3090 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, as a modelling tool in applied mathematics.

### 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
• Explain the relation between some network structural and spectral properties, and their significance for real-world networks
• Utilise graph theoretical tools to determine the stability of complex dynamical systems
• Explain real world phenomena in complex dynamical networks, such as synchronisation and scale-free network structures
• Express stochastic processes mathematically as dynamical equations on networks for the probability of stochastic outcomes.
• Extract information about stochastic processes, such as the probability distribution and critical phenomena.

### 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 - Dynamics on complex networks: stability and asymptotic trajectories - Examples of dynamical processes on and of networks: models of coupled oscillators, growing (scale-free) networks Part III: Stochastic dynamics on networks - Introduction to stochastic processes and how they related to dynamical systems on networks - Discussion of particular real world stochastic systems: random walks, gene regulatory networks, epidemics on social 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
Teaching48
Independent Study102
Total study time150

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

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

### Assessment

#### Summative

MethodPercentage contribution
In-class Test  (40 minutes) 33.33%
In-class Test  (40 minutes) 33.34%
In-class Test  (40 minutes) 33.33%

#### Referral

MethodPercentage contribution
Final Exam 100%

#### Repeat Information

Repeat type: Internal & External