  We analyze the critical connectivity of systems of penetrable $d$-dimensional
spheres having size distributions in terms of weighed random geometrical
graphs, in which vertex coordinates correspond to random positions of the
sphere centers and edges are formed between any two overlapping spheres. Edge
weights naturally arise from the different radii of two overlapping spheres.
For the case in which the spheres have bounded size distributions, we show that
clusters of connected spheres are tree-like for $d\rightarrow \infty$ and they
contain no closed loops. In this case, we find that the mean cluster size
diverges at the percolation threshold density $\eta_c\rightarrow 2^{-d}$,
independently of the particular size distribution. We also show that the mean
number of overlaps for a particle at criticality $z_c$ is smaller than unity,
while $z_c\rightarrow 1$ only for spheres with fixed radii. We explain these
features by showing that in the large dimensionality limit the critical
connectivity is dominated by the spheres with the largest size. Assuming that
closed loops can be neglected also for unbounded radii distributions, we find
that the asymptotic critical threshold for systems of spheres with radii
following a lognormal distribution is no longer universal, and that it can be
smaller than $2^{-d}$ for $d\rightarrow\infty$.
