is always assumed to be an arbitrary fixed integer. , $B_r(\xi)=\{\eta\in\R^d:\,|\eta-\xi|\le r\}$ denotes the ball of radius $r>0$ centered at $\xi\in\R^d$.

Scale-free Gilbert graph {#htggSec}
------------------------

Let $X^{(n)}$ be an independently $[0,\infty)$-marked homogeneous Poisson point process with intensity $1$ in the torus $\T_n$, where $\T_n$ is obtained from the cube $[-n/2,n/2]^d$ by the standard identification of its boundaries. If $n\ge1$, $r>0$ and $\xi\in\T_n$, then we write $\BT_r(\xi)=\{\eta\in\T_n:\d_{\T_n}(\xi,\eta)\le r\}$ for the closed ball in $\T_n$ with radius $r$ centered at $\xi$, where $\d_{\T_n}(\xi,\eta)$ denotes the toroidal distance between $\xi$ and $\eta$. The mark of a point from $X^{(n)}$ is interpreted as the radius of a ball centered at this point. Throughout the paper, we assume that the distribution of the typical mark $R$ is absolutely continuous and heavy-tailed. That is, $\beta,s\in(0,\infty)$ $\lim_{h\to\infty} h^{s}\P(R>h)=\beta$. For every $\varepsilon\in(0,1)$, we also fix $t_0=t_0(\varepsilon)>0$ $t^{s}\P(R>t)\in (\beta(1-\varepsilon),\beta(1+\varepsilon))$ $t\ge t_0(\varepsilon)$.

For $n\ge1$ we investigate the directed random geometric graph $G_{}(X^{(n)})$ on the vertex set $X$, where an edge in $G(X^{(n)})$ is drawn from $x=(\xi,r)\in X^{(n)}$ to $y=(\eta,t)\in X^{(n)}$ if $\eta\in\BT_r(\xi)$. First, we investigate the asymptotic distributions of the power-weighted sum of all outgoing and all incoming edge lengths. To be more precise, fix $\alpha\ge0$, $n\ge1$ and let 