this configuration. In order to reduce the total network length, we therefore introduce a variant $G^\p$ of the original graph $G$, where such redundancies are removed.

(0,0) circle (3cm); (1.4,1.0) circle (1.5cm); (2.5,0.2) circle (1cm);

(0,0) circle (3cm); (1.4,1.0) circle (1.5cm); (2.5,0.2) circle (1cm);

(0,0) circle (2pt); (1.4,1.0) circle (2pt); (2.5,0.2) circle (2pt);

(0,0) –(1.4,1.0)–(2.5,0.2); (0,0) – (2.5,0.2);

\(u) at (0.0,0.0); (u) at (1.4,1.2); (u) at (2.4,0.0);

To be more precise, for any finite subset $\varphi$ of $\T_n\times[0,\infty)$ define the *thinned scale-free Gilbert graph* $G^\p(\varphi)$ as the graph on $\varphi$, where an edge is drawn from $(\xi,r)\in\varphi$ to $(\eta,t)\in\varphi$ if $(\eta,t)\in \BT_r(\xi)\times[0,r)$ and there does not exist $(\zeta,w)\in\varphi\cap (\BT_r(\xi)\times(t,r))$ $\eta\in\BT_w(\zeta)$.

By definition, $G^\p(X^{(n)})$ is a subgraph of $G(X^{(n)})$ and we will see in Proposition \[conProp\] that the thinning does not affect the connected components of $G(X^{(n)})$. From the point of view of telecommunication networks, we can reach the same set of subscribers using a smaller cable length. Next, we investigate the question whether the reduction of cable length is substantial. For concreteness, we assume that $s=d$. We show that the leading order of the expected power-weighted sum of lengths of outgoing edges in $G^\p(\{(o,R^*)\}\cup X^{(n)})$ is strictly smaller when compared 