Erdős–Rényi Phase Transition

Emergence of the giant connected component at <k> = 1

Mean degree <k>
Giant component / N
Num. components
Edges
80
0.80
In G(N, p) random graphs, a phase transition occurs at ⟨k⟩ = 1 (p = 1/N). Below: only small tree-like clusters. At ⟨k⟩ = 1: a giant component containing O(N) nodes appears. Above: the giant component rapidly absorbs smaller clusters. The giant fraction S satisfies the self-consistency equation S = 1 − e^{−⟨k⟩S}. Node size/color encodes component membership — the giant cluster glows lime green.