asymptotic behavior differs decisively from the one of either long-range percolation [@lrp1; @lrp2] or ultra-small scale-free geometric networks [@ultraSmall]. As we will see in the proof of Theorem \[sd1Thm\], this scaling regime is a consequence of the close relationship between scale-free Gilbert graphs and fractal percolation processes.

\[sd1Thm\] Assume that $\beta>d^{d/2}2^{2d+1}(d+1)\log 2$. Then, the graph $G(X^{(n)})$ is connected whp, and, moreover, the diameter $\mathsf{diam}\,G(X^{(n)})$ of $G(X^{(n)})$ is stochastically dominated by an affine function in the total progeny of a subcritical Galton-Watson process whp. To be more precise, a coupling between $X^{(n)}$ and the total progeny $T$ of a subcritical Galton-Watson process $\P(\mathsf{diam}\,G(X^{(n)})>2+2^dT)\in O(n^{-1})$.

Thinned scale-free Gilbert graph {#thtggSec}
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As explained in Section \[intSec\], the scale-free Gilbert graph $G(X^{(n)})$ contains a large number of redundant edges, which can be removed without affecting its connected components. To be more precise, if $x_1=(\xi_1,r_1),x_2=(\xi_2,r_2),x_3=(\xi_3,r_3)\in X^{(n)}$ are $r_1>r_2>r_3$, $\xi_2\in \BT_{r_1}(\xi_1)$ and $\xi_3\in \BT_{r_1}(\xi_1)\cap \BT_{r_2}(\xi_2)$, then in $G(X^{(n)})$ the point $x_3$ is connected both to $x_1$ and $x_2$. However, the edge from $x_3$ to $x_1$ is redundant since one can also reach $x_1$ from $x_3$ by first moving from $x_3$ to $x_2$ and then from $x_2$ to $x_1$. See Figure \[thinCon\] for an illustration of 