🕸️ Erdős–Rényi Percolation Lab

Graph G(N, p) Nodes N 80 Mean degree ⟨c⟩ = p(N−1) 0.50 Random seed 1

Sweep speed 0.25

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Live readouts

Edge probability p— Edges (measured / expected)— Measured ⟨k⟩— Components— Giant component— Giant fraction S (measured)— Giant fraction S (theory, N→∞)— Mean finite-cluster size ⟨s⟩— Isolated nodes (meas. / N·e^−c)— Frame rate—

Graphs

Giant fraction vs ⟨c⟩ — line: theory S = 1 − e^−cS, dots: this graph

Degree distribution — bars: graph, dots: Poisson(⟨c⟩)

Independent numeric checks

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The science

G(N,p): each of the N(N−1)/2 pairs is linked independently with probability p. Mean degree ⟨c⟩ = p(N−1).

Degrees are Binomial(N−1,p), which tends to Poisson: P(k) = e^−c c^k/k!.

Giant component: its fraction S obeys S = 1 − e^−cS. S = 0 for c ≤ 1; just above threshold S ≈ 2(c−1). At c = 1 the largest cluster scales as N^2/3.

Below threshold the mean size of the cluster containing a random node is ⟨s⟩ = 1/(1−c); it diverges at c = 1 (finite N rounds it off).

Full connectivity needs c ≈ ln N: the expected number of isolated nodes is N e^−c.

Edges use a fixed random number per pair (edge present if U_ij < p), so raising ⟨c⟩ only adds edges — a true monotone percolation process. Node positions come from a Fruchterman–Reingold force layout and carry no meaning. Simplifications: the theory is for N→∞ while the sim has finite N, so deviations near c = 1 are expected.