Erdős-Rényi Random Graph G(n,p)

Phase transition: At c = np = 1 (critical point), a giant component suddenly appears containing O(n^(2/3)) vertices. For c < 1: largest component ~ O(log n). For c > 1: giant component ~ β·n where β satisfies β = 1 − e^−cβ (implicit equation).
Component size distribution: at c=1, P(component size = k) ~ k^-3/2 (power law with exponent 3/2). Percolation on random graphs is equivalent to bond percolation on the complete graph. Giant shown in green; other components in gray.