  Define the scale-free Gilbert graph based on a Boolean model with
heavy-tailed radius distribution on the $d$-dimensional torus by connecting two
centers of balls by an edge if at least one of the balls contains the center of
the other. We investigate two asymptotic properties of this graph as the size
of the torus tends to infinity. First, we determine the tail index associated
with the asymptotic distribution of the sum of all power-weighted incoming and
outgoing edge lengths at a randomly chosen vertex. Second, we study the
behavior of chemical distances on scale-free Gilbert graphs and show the
existence of different regimes depending on the tail index of the radius
distribution. Despite some similarities to long-range percolation and
ultra-small scale-free geometric networks, scale-free Gilbert graphs are
actually more closely related to fractal percolation and this connection gives
rise to different scaling limits. We also propose a modification of the graph,
where the total number of edges can be reduced substantially at the cost of
introducing a logarithmic factor in the chemical distances.
