Sparse Random Graphs 1

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Erdos–Rényi G(n,p) with p=λ/n exhibits a sharp phase transition at λ=1: λ<1 gives only microscopic components (expected size ≤1/(1−λ), max ~log n), while λ>1 yields a unique giant of fraction ζ(λ) where 1−ζ = exp(−λζ). Proof uses breadth‑first exploration coupled to a Poisson branching process coupling, controls short‑cycle “conflicts” and grows boundaries to ~√n to force intersections; typical graph distances scale like (½)·log n per side, so total ~log n.

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1. 0:00Start
2. 1:15Sparse Random Graphs
3. 7:17Model of Random Graph
4. 10:28Phase Transition
5. 12:21Breadth First Exploration
6. 22:54Branching Process
7. 31:39Proportion of Vertices in Components of Size Greater than Equal to L
8. 1:18:56Local Neighborhood Approximation