Math

The birth of the giant component

Add one edge, find a path between two vertices, then watch c pass through one. The graph and all readings use the same edge stream.

By Mykhailo Makarov

Actual application image

What happens

Add one edge, find a path between two vertices, then watch c pass through one. The graph and all readings use the same edge stream. Uniform random unordered pairs of distinct vertices, sampled with replacement. Loops are excluded; repeated pairs count in E. This is a multigraph, not a simple G(N,E) sample. The curve S = 1 − exp(−cS) is a sparse large-N limit: c = 2E/N, so c = 1 means E ≈ N/2. A finite graph has no prescribed transition edge. Real networks need their own models.

Equations used in the module

Equations
c = 2E/N
S = 0 (c ≤ 1); S = 1 − exp(−cS) (c > 1), N → ∞
L₁/N; L₂/N
d(A,B) = min |P(A,B)|

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