Induced subgraphs in sparse random graphs with given degree sequence

For any $S\subset [n]$, we compute the probability that the subgraph of $\mathcal{G}_{n,d}$ induced by $S$ is a given graph $H$ on the vertex set $S$. The result holds for any $d=o(n^{1/3})$ and is further extended to $\mathcal{G}_{\bf d}$, the probability space of random graphs with a given degree sequence $\bf d$.

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