  We study networks in $\R^n$ which are periodic under a lattice of rank~$n$
and have vertices of prescribed degree $d\ge 3$. We minimize the length of the
quotient networks, subject to the constraint that the fundamental domain has
$n$-dimensional volume~$1$. For $n=3$ and degree $3\leq d\leq 6$ we determine
the minimizing networks with the least number of vertices in the quotient,
while for $d\ge 7$ we state a length estimate. For general $n$, we determine
the unique minimizers with $d=n+1$ and $d=2n$.
