under suitable assumptions on the tail behavior of the radius distribution, we show in Theorem \[redElThm\] that the sum of power-weighted edge lengths at a randomly picked vertex decreases significantly. Of course, reducing edge lengths comes at the price of longer connection paths, but we will show in Theorem \[intMedProp\] that chemical distances grow at most by a logarithmic factor in the size of the torus.

The present paper is organized as follows. In Section \[defSec\], we provide precise definitions of the scale-free and thinned scale-free Gilbert graphs and state our main results. Next, in Section \[sfSec\], we investigate power-weighted sums of edge lengths at a typical point and prove Theorems \[outEdgeThm\], \[inEdgeThm\] and \[asyGrowProp\]. Section \[chemDistSec\] is devoted to the analysis of chemical distances for different regimes of the tail index. Finally, in Section \[multiSec\], we investigate the changes in terms of the asymptotic behavior of power-weighted sums of edge lengths and of chemical distances as we move from the scale-free Gilbert graph to its thinning.

Model definition and statement of main results {#defSec}
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In Sections \[htggSec\] and \[thtggSec\], we provide precise definitions of the network models under consideration and present our main results. In the following, $d\ge2$ 