 Isolated vertices in the Erdős-Rényi model

ID: isolated-vertices-in-the-erdos-renyi-model

Codex  0 Created 2026-09-24 Updated 2026-09-28

Let

N

G (n, p)

. Indicator variables give

E N = n (1 - p)^{n - 1}

and, because a specified pair is simultaneously isolated precisely when the

2 n - 3

incident edges are absent,

E (N^{2}) = n (1 - p)^{n - 1} + n (n - 1) (1 - p)^{2 n - 3} .

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