we investigate the asymptotic distribution of the power-weighted sum of all incoming and all outgoing edge lengths considered from a vertex that is picked uniformly at random, see Theorems \[outEdgeThm\] and \[inEdgeThm\]. In particular, our results imply that the asymptotic out-degree and in-degree distributions admit polynomial tails. We also investigate the growth of the expected power-weighted sum of all outgoing edge lengths as $n\to\infty$, see Theorem \[asyGrowProp\]. In Section \[chemDistSec\], we show that different scaling regimes of chemical distances emerge depending on the tail index of the radius distribution.

Since the scale-free Gilbert graph exhibits a large degree of redundancy of connections, in Section \[multiSec\] we present a variant, called *thinned scale-free Gilbert graph*, that substantially decreases these redundancies by introducing a multi-layer topology. Loosely speaking, an edge in the original graph from a low-level node $x$ to a high-level node $y$ is removed if $y$ can also be reached from $x$ via an intermediate node $z$. See Figure \[onnFig\] for an illustration of the thinned scale-free Gilbert graph. A precise definition of this graph will be given in Section \[defSec\], but we note already at this point that removing redundancies does not alter the family of connected components. , 