Erdős–Rényi Phase Transition

G(n,p): place n nodes, connect each pair with probability p. At c=np=1, a giant connected component suddenly emerges — a sharp phase transition analogous to percolation.

Parameters

Nodes n 80

Mean degree c = np 1.00

Nodes n80

Edge prob p—

Edges—

Giant comp.—

GC fraction—

Components—

—

c<1: only small trees
c=1: critical point
c>1: giant component

GC size ~ 2(c-1)/c²·n
(Erdős-Rényi 1960)