General
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- 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
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