Erdős-Rényi Giant Component Threshold

G(n,p) random graph: percolation transition at ⟨k⟩ = np = 1

Graph Parameters
Phase Diagram
State
Giant comp. S: —
S/N (frac): —
Components: —
Theory S/N: —

Erdős-Rényi Threshold Phenomenon

In G(n,p) with p=c/n, Erdős and Rényi (1960) proved a sharp threshold at c=1: for c<1, all components have size O(log n). For c>1, a unique "giant component" emerges containing a fraction β(c) of all nodes, where β satisfies β = 1 − e^(−c·β) (self-consistency equation).

Near the threshold c=1±ε, the giant component grows as β ~ 2ε (linear onset). The second-largest component stays O(n^(2/3)) — this is analogous to critical percolation on a bethe lattice with coordination number c.

The "sweep" button builds the phase diagram by computing giant component fraction vs. c, comparing to the theoretical curve from the implicit equation.