in $n$. To be more precise, putting $\mathsf{e}_1=(1,0,\ldots,0)\in\T_n$ and denoting by $q(x)$ the closest point of $X$ seen from a given point $x\in\T_n$, we have the following result.

\[subCrit\] Assume that $s>d$ and let $\alpha>0$ be arbitrary. Then, the chemical distance between $q(-n\mathsf{e}_1/4)$ and $q(n\mathsf{e}_1/4)$ is at least $n/(\log n)^{\alpha}$ whp.

We conjecture that the sublogarithmic correction factor is only an artifact of our proof.

Assume that $s>d$. Then, there exists a constant $c=c(\beta,d)>0$ the chemical distance between $q(-n\mathsf{e}_1/4)$ and $q(n\mathsf{e}_1/4)$ is at least $cn$ whp.

Finally, we assume that $s=d$. On the one hand, in contrast to the case $s>d$ the heavy-tailedness of the radius distribution has a substantial effect on chemical distances. On the other hand, the connectivity structure still does not degenerate as in the case $s<d$. Therefore, from the point of view of modeling scale-free network structures, the regime $s=d$ might be considered to be the most interesting one. In Theorem \[sd1Thm\] below, we show that if the parameter $\beta$ is sufficiently large, then not only the graph $G(X^{(n)})$ is connected whp, but, moreover, its diameter remains bounded in the sense that whp it is dominated by the total progeny of a subcritical Galton-Watson process. This 