Oxford Mathematics Lectures: “Networks (Renaud Lambiotte)”, “Riemannian Geometry (Jason Lotay)”
This is a good lecture on network science (YouTube playlist) by Renaud Lambiotte from Oxford Mathematics (YouTube Channel).
Lambiotte, R. and Masuda, N., 2021. A guide to temporal networks. World Scientific.
Course archive C5.4 Networks, materials download:
- Introduction and short overview of useful mathematical concepts (2 lectures): Networks as abstractions; Renewal processes; Random walks and diffusion; Power-law distributions; Matrix algebra; Markov chains; Branching processes.
- Basic structural properties of networks (2 lectures): Definition; Degree distribution; Measures derived from walks and paths; Clustering coefficient; Centrality Measures; Spectral properties.
- Models of networks (2 lectures): Erdos-Rényi random graph; Configuration model; Network motifs; Growing network with preferential attachment.
- Community detection (2 lectures): Newman-Girvan Modularity; Spectral optimization of modularity; Greedy optimization of modularity.
- Dynamics, time-scales and Communities (2 lectures): Consensus dynamics; Timescale separation in dynamical systems and networks; Dynamically invariant subspaces and externally equitable partitions
- Dynamics I: Random walks (2 lectures): Discrete-time random walks on networks; PageRank; Mean first-passage and recurrence times; Respondent-driven sampling; Continous-Time Random Walks
- Random walks to reveal network structure (2 lectures): Markov stability; Infomap; Walktrap; Core–periphery structure; Similarity measures and kernels
- Dynamics II: Epidemic processes (2 lectures): Models of epidemic processes; Mean-Field Theories and Pair Approximations
Another related lecture is on Riemannian geometry by Jason Lotay from Oxford Mathematics
Course archive C3.11 Riemannian Geometry
The lectures are not https://www.youtube.com/watch?v=wZgM3u8UkNs&list=PL4d5ZtfQonW17IBjdLKcfQVBuuKaWnxbx
- Riemannian manifolds: basic examples of Riemannian metrics, Levi-Civita connection.
- Geodesics: definition, first variation formula, exponential map, minimizing properties of geodesics.
- Curvature: Riemann curvature tensor, sectional curvature, Ricci curvature, scalar curvature.
- Riemannian submanifolds: examples, second fundamental form, Gauss–Codazzi equations.
- Jacobi fields: Jacobi equation, conjugate points.
- Completeness: Hopf–Rinow and Cartan–Hadamard theorems
- Constant curvature: classification of complete manifolds with constant curvature.
- Second variation and applications: second variation formula, Bonnet–Myers and Synge’s theorems.