Course: C5.4 Networks (2024-25) | Mathematical Institute

Lecturer: Profile: Peter Grindrod

Course information

General Prerequisites:

Basic notions of linear algebra, probability theory, and some computational experience. Numerical codes may be illustrated in tutorials, but the student has the possibility to use the language of their choice. Relevant notions of graph theory will be reviewed.

Course Term: Hilary

Course Lecture Information: 16 lectures

Course Weight: 1

Course Level: M

Course Overview:

Network Science provides generic tools to model and analyse systems in a broad range of disciplines, including biology, computer science and sociology. There are commercial applications within media marketing, defence and security, ranking and seeking in sports, and many sectors of the digital economy. This course aims at providing an introduction to this interdisciplinary field of research, by integrating tools from graph theory, statistics, linear algebra, and dynamical systems. Most of the topics to be considered are active modern research areas.

Learning Outcomes:

Students will have developed a sound knowledge and appreciation of some of the tools, concepts, models, and computations used in the study of networks. The study of networks is predominantly a modern subject, so the students will also be expected to develop the ability to read and understand current (2015-2025) research papers in the field.

Course Synopsis:

1. Introduction and short overview of useful mathematical concepts : Networks as abstractions; Renewal processes; Random walks and diffusion; Power-law distributions; Matrix algebra; Markov chains; Branching processes.

2. Basic structural properties of networks : Definition; Degree distribution; Measures derived from walks and paths; Clustering coefficient; Centrality Measures; The Laplacian; Spectral properties.

3. Models of networks : Erdos-Rényi random graph; Configuration model; Small World graphs, Network motifs; Core–periphery structure; Growing network with preferential attachment.

4. Community detection (2 lectures): Newman-Girvan Modularity; Spectral optimization of modularity; Greedy optimization of modularity.

5. Dynamics, Time-scales and Communities; Consensus dynamics; Timescale separation in dynamical systems and networks; Dynamically invariant subspaces and externally equitable partitions

6. Dynamics: Random walks Discrete-time random walks on networks; PageRank; Mean first-passage and recurrence times; Respondent-driven sampling; Continuous-Time Random Walks, Models of epidemic processes

7: Scaling Laws for functionals of growing networks

8. Dynamics:of networks Birth and dearth of edges: Stochastic models, Mean-Field Theories and Approximations

9. Applications to large scale simulations of the human cortex

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- Select activity 10 UK cities' reciprocated mentions networks (giant components)
  
  10 UK cities' reciprocated mentions networks (giant components) File
  
  This is data wit reciprocated mentions taken from https://royalsocietypublishing.org/doi/10.1098/rsos.150526
  
  We defined a reciprocated (Twitter) mentions network, over 10 UK cities and a four week period, as follows. We identified all those Twitter accounts that were active during 1–28 October 2014 (four full weeks), having biographical information indicating that they are associated with one of 10 cities: London, Manchester, Leeds, Sheffield, Nottingham, Birmingham, Bristol, Cardiff, Glasgow and Edinburgh. The data were collected, using the DataSift Twitter Firehose access, by selecting individuals who had one of the 10 cities stated in their user description field. (We did not look for geo-coded Tweets because the coverage would be too sparse for it to be useful.) We trapped all Tweets during that period (excluding all ReTweets, which accounted for around one-third of all Tweets). This resulted in 14 946 171 original Tweets made by 321 255 individual accounts.
- Select activity Problem Sheet 2 and 2a(NBTW) (Taken together)
  
  Problem Sheet 2 and 2a(NBTW) (Taken together) Assignment
  
  Opened: Thursday, 17 October 2024, 12:00 AM
- Select activity Problem Sheet 1
  
  Problem Sheet 1 Assignment
  
  Opened: Thursday, 17 October 2024, 12:00 AM
  
  Due: Thursday, 14 November 2024, 9:52 AM
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  Problem Sheet 3 Assignment
  
  Opened: Wednesday, 6 November 2024, 12:00 AM
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  Problem Sheet 4 Assignment
  
  Opened: Saturday, 23 November 2024, 12:00 AM
- Select activity Prob seheet 2 solutions to 2 and 3
  
  Prob seheet 2 solutions to 2 and 3 Assignment
  
  Opened: Wednesday, 6 November 2024, 12:00 AM
  
  Due: Wednesday, 13 November 2024, 12:00 AM
- Select activity Fly files: Central region in the Drosophila connecome
  
  Fly files: Central region in the Drosophila connecome
  
  These are files with the nodes (neurons) and all pairwise directed edges (all synaptic connections) identified between the neurons within the Cntral (processing) part of the Drosophila fruit fly brain, published in 2024.
  
  These files were used in the draft paper https://arxiv.org/pdf/2412.13202 . See the references for the original data there and here, by the Fly- Wire collaboration https://codex.flywire.ai .
- Select activity More fly data
  
  More fly data File
- Select activity 2025 Update of course notes
  
  2025 Update of course notes File
- Select activity Lecture 9 slides on "Dynamics on Networks", as an experiment
  
  Lecture 9 slides on "Dynamics on Networks", as an experiment File
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  Registration start: Monday, 13 January 2025, 12:00 PM
  
  Registration end: Friday, 14 February 2025, 12:00 PM
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