Giant Component Phase Transition

In an Erdős–Rényi graph G(N,p), each pair of N nodes is connected independently with probability p. As p increases past the critical threshold p_c = 1/N (mean degree ⟨k⟩ = 1), a dramatic phase transition occurs: a giant connected component suddenly spans a finite fraction of all nodes.

Theory (Erdős & Rényi, 1959-60): For ⟨k⟩ = c = pN, the giant component fraction S satisfies: S = 1 − e^(−cS) — a self-consistency equation. For c<1: S=0. For c>1: S>0 emerges continuously.