  Let $[\mathcal{P}]$ be the points of a Poisson process on $\mathbb{R}^d$ and
$F$ a probability distribution with support on the non-negative integers.
Models are formulated for generating translation invariant random graphs with
vertex set $[\mathcal{P}]$ and iid vertex degrees with distribution $F$, and
the length of the edges is analyzed. The main result is that finite mean for
the total edge length per vertex is possible if and only if $F$ has finite
moment of order $(d+1)/d$.
