Connectivity of inhomogeneous random graphs II

Each graphon $W:\Omega^2\rightarrow[0,1]$ yields an inhomogeneous random graph model $G(n,W)$. We show that $G(n,W)$ is asymptotically almost surely connected if and only if (i) $W$ is a connected graphon and (ii) the measure of elements of $\Omega$ of $W$-degree less than $\alpha$ is $o(\alpha)$ as $\alpha\rightarrow 0$. These two conditions encapsulate the absence of several linear-sized components, and of isolated vertices, respectively.
We study in bigger detail the limit probability of the property that $G(n,W)$ contains an isolated vertex, and, more generally, the limit distribution of the minimum degree of $G(n,W)$.