Hub paths and the largest component of the barely subcritical configuration model

For the configuration model with a degree sequence whose third moment converges, write $\varepsilon_n=1-\nu_n$, where $\nu_n\to1$ from below is the size-biased mean minus one. Van der Hofstad asked to show that in the barely subcritical regime the largest component has order $\max\{d_{\max}/\varepsilon_n, \varepsilon_n^{-2}\log(n\varepsilon_n^3)\}$, and predicted that it has size $(d_{\max}/\varepsilon_n)(1+o_\mathbb{P}(1))$ when $\varepsilon_n$ tends to zero sufficiently slowly. We show that neither holds in this generality. We use degree sequences with vertices of degree one and three and a polynomial number of vertices of a large common degree $d$. Paths of $K$ such hubs joined by direct edges then appear, and the component of such a path is a tree with about $Kd/\varepsilon_n$ vertices. Taking $K=2$ gives $L_1\ge(2-\eta)d_{\max}/\varepsilon_n$ with high probability, for $\varepsilon_n$ tending to zero as slowly as we like; letting $K\to\infty$ gives $L_1/\max\{d_{\max}/\varepsilon_n,\varepsilon_n^{-2}\log(n\varepsilon_n^3)\}\to\infty$ in probability. In all examples the degree of a uniform vertex converges in law to a variable on $\{1,3\}$ with all three moments, and $n\varepsilon_n^3\to\infty$.