ERDŐS-RÉNYI RANDOM GRAPH G(n,p)

Giant Component Emergence · p_c = 1/n · Phase Transition

PARAMETERS

p = c/n = 0.0125

GRAPH STATS

Nodes n: 80
Edges m: 0
Giant comp S: 0
S/n: 0.000
⟨k⟩ = c: 1.00
Threshold:
p_c = 1/n (c_c = 1)
S/n → 1-e^(-cS/n)

ORDER PARAMETER

S/n vs c — giant component

ABOUT

In G(n,p) with p=c/n, a giant component of size ~S/n emerges at c=1. Below: all components O(log n). Above: one component O(n). The order parameter S/n obeys S=1-e^(-cS) — mean-field self-consistency. Erdős & Rényi 1960.
Top: graph (giant component green)   Bottom: degree distribution vs Poisson(c)