Giant Component Emergence — Erdős–Rényi G(n,p)

In an Erdős–Rényi random graph G(n,p), as the mean degree ⟨k⟩ = p(n-1) crosses 1, a giant connected component suddenly emerges containing O(n) vertices — a phase transition analogous to percolation. Below the threshold, components are O(log n); above it, a single giant component dominates.