The Erdős–Rényi model G(n, p) connects every pair of n
nodes independently with probability p. As p rises,
something surprising happens: instead of the graph growing gradually more
connected, it undergoes a sharp phase transition. Below a
threshold the graph is a scatter of small, isolated fragments. Above it,
almost all of a sudden, a single giant component swallows a
finite fraction of all the nodes.
⟨c⟩ = p(n−1) = 1 — below it, components stay logarithmically small; above it, one component grows linearly with n.p to decide whether an edge is drawn.⟨c⟩ = 1.p = c/(N−1); drag slowly through ⟨c⟩ = 1 to see the transition.
Erdős and Rényi's 1959–60 papers founded random graph theory. The same
⟨c⟩ = 1 threshold shows up in bond percolation on lattices,
the spread of epidemics (the basic reproduction number R₀),
and the reliability of communication networks — anywhere connections form
at random, a giant component appears with startling suddenness.
A 3D random graph of N nodes where every possible edge is drawn independently with probability p, letting you watch the network snap from scattered fragments into one dominant giant component as the average degree crosses 1.
Erdős–Rényi random graphs undergo a sharp phase transition at average degree ⟨c⟩ = 1: below it, components stay small and numerous; above it, a single component absorbs a finite fraction of all nodes almost overnight.
Set the node count and drag the average-degree slider slowly through ⟨c⟩ = 1 while watching the highlighted giant component and live component statistics. Reroll to see a fresh random draw at the same settings.
The same ⟨c⟩ = 1 threshold governs epidemic outbreak size (basic reproduction number R₀), lattice bond percolation, and the robustness of real communication networks to random failures.