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).

Reference book:
Lambiotte, R. and Masuda, N., 2021. A guide to temporal networks. World Scientific.

Course archive C5.4 Networks, materials download:

Course Synopsis:

1. 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.
2. Basic structural properties of networks (2 lectures): Definition; Degree distribution; Measures derived from walks and paths; Clustering coefficient; Centrality Measures; Spectral properties.
3. Models of networks (2 lectures): Erdos-Rényi random graph; Configuration model; Network motifs; 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 (2 lectures): Consensus dynamics; Timescale separation in dynamical systems and networks; Dynamically invariant subspaces and externally equitable partitions
6. 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
7. Random walks to reveal network structure (2 lectures): Markov stability; Infomap; Walktrap; Core–periphery structure; Similarity measures and kernels
8. 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

1. Riemannian manifolds: basic examples of Riemannian metrics, Levi-Civita connection.
2. Geodesics: definition, first variation formula, exponential map, minimizing properties of geodesics.
3. Curvature: Riemann curvature tensor, sectional curvature, Ricci curvature, scalar curvature.
4. Riemannian submanifolds: examples, second fundamental form, Gauss–Codazzi equations.
5. Jacobi fields: Jacobi equation, conjugate points.
6. Completeness: Hopf–Rinow and Cartan–Hadamard theorems
7. Constant curvature: classification of complete manifolds with constant curvature.
8. Second variation and applications: second variation formula, Bonnet–Myers and Synge’s theorems.