Sparse Random Graphs 2

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Survey of sparse random graphs ties models to concrete thresholds and methods: stochastic block graphs reduce locally to multi-type branching with a giant iff the Perron eigenvalue of the mean matrix exceeds 1; configuration model neighborhoods follow a size-biased distribution and give a giant when nu = E[D(D−1)]/E[D] > 1; basic preferential attachment produces a degree tail ~ k−3; percolation couples to SIR epidemics and satisfies E|cluster| ≤ √n/(1−pλ1(A)); component sizes at criticality rescale as n2/3 via exploration encoded by a walk and proved with a martingale FCLT.

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1. 0:00Start
2. 2:11Local Neighborhood Approximation
3. 5:37Stochastic Block Model
4. 7:33Local Neighborhood Approximation of Stochastic Block Model
5. 8:32Neighborhood Distribution
6. 10:23Multi-Type Branching Processes
7. 16:04Configuration Model
8. 21:30Convergence of Moments
9. 31:57Size Biased Distribution
10. 38:35Dynamically Growing Networks
11. 43:40Degree Distribution
12. 46:49Percolation and Epidemics on Networks
13. 47:33Percolation Threshold
14. 55:24The Percolation Threshold
15. 1:21:28Summary