D^{(\alpha)}_{\inn,n}=\beta  \int_{[-1/2,1/2]^d}|\eta|^{\alpha-s}\d \eta$.

Next, we investigate the asymptotic behavior of chemical distances in scale-free Gilbert graphs, where we allow a directed edge in $G(X^{(n)})$ to be traversed in both directions. For the analysis of $G(X^{(n)})$, it will be convenient to distinguish between the regimes $s<d$, $s=d$ and $s>d$. We say that a family of events occurs *with high probability (whp)* if the probabilities of the events tend to $1$ as $n\to\infty$. In the case where $s<d$, whp there exists some point of $X^{(n)}$ that is connected to all other points by an edge in $G(X^{(n)})$. , the diameter of $G(X^{(n)})$ is at most $2$ whp. The cases $s=d$ and $s>d$ are more subtle. If $s>d$, then as $n\to\infty$ the probability that $G(X^{(n)})$ contains isolated vertices is bounded away from $0$, see Proposition \[nonConnLem\]. Still, if $\beta$ is large, then it follows from continuum percolation that whp there exists a giant component containing a positive proportion of the vertices. But even inside this giant connected component the effects of the heavy-tailed nature of $R$ are barely noticeable. This is to be understood in the sense that chemical distances (i.e., minimal number of hops) between nodes at distance $n$ grow almost linearly 