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.
Prerequisites: MATH1048 and MATH1049 and MATH1059
Aims and Objectives
Having successfully completed this module you will be able to:
- Utilise graph theoretical tools to determine the stability of complex dynamical systems
- Explain the relation between some network structural and spectral properties, and their significance for real-world networks
- Express stochastic processes mathematically as dynamical equations on networks for the probability of stochastic outcomes.
- Define basic network properties, compute them in theoretical and practical situations, and explain their significance in network modelling
- Extract information about stochastic processes, such as the probability distribution and critical phenomena.
- Explain real world phenomena in complex dynamical networks, such as synchronisation and scale-free network structures
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.
|Total study time||150|
Resources & Reading list
M.E.J. Newman (2010). Networks: An Introduction. Oxford University Press.
A.-L. Barabasi (2016). Network Science. Cambridge University Press.
This is how we’ll formally assess what you have learned in this module.
This is how we’ll assess you if you don’t meet the criteria to pass this module.
Repeat type: Internal & External