Erdős-Rényi Graph — Giant Component Threshold

About: In G(n,p), edges appear independently with probability p. Erdős and Rényi (1960) proved a sharp phase transition: when ⟨k⟩ = np < 1, all components are small (O(log n)); when ⟨k⟩ > 1, a giant component of size Θ(n) emerges. At ⟨k⟩ = 1, the largest component has size O(n^(2/3)). The right panel plots measured giant component fraction vs ⟨k⟩ accumulated across samples.