On Universal Representation of Random Graphs

Abstract

It is shown that every probability measure μ on the interval [0, 1] gives rise to a unique infinite random graph g on vertices {v ₁ , v ₂ , . . .} and a sequence of random graphs gn on vertices {v ₁ , . . . , v _n } such that $\mu (g_n \rightarrow g)$. In particular, $\mathbf{P}(G-n(Q))$ for Bernoulli graphs with stable property Q, can be strengthened to: ∃ probability space (Ω, F, P), ∃ set of infinite graphs G(Q) ∈, F with property Q such that $P (G_n(Q) \rightarrow G(Q)) = 1 \quad \mathrm{and}\quad \mathbf{P}(G-n(Q)) = P(G_n(Q))$.