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Let $D_n$ be the degree of a random vertex. We let $\\nu_n={\\mathbb E} [D_n(D_n-1)]/{\\mathbb E}[D_n]$ and, assuming that $\\nu_n \\to 1$ as $n \\to \\infty$, we write $\\varepsilon_n=\\nu_n-1$. We call the setting where $\\varepsilon_n n^{1/3}/({\\mathbb E}[D_n^3])^{2/3} \\to \\infty$ the {\\it barely supercritical} regime. We further assume that the variance of $D_n$ is uniformly bounded as $n \\to \\infty$.\n  Let $D_n^*$ denote the size-biased version of $D_n$. 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