  The general linear group $GL_{n}$, along with its adjoint simple group
$PGL_n$, can be described by means of weighted planar networks. In this paper
we give a network description for simple Lie groups of types $B$ and $C$. The
corresponding networks are axially symmetric modulo a sequence of cluster
mutations along the axis of symmetry. We extend to this setting the result of
Gekhtman, Shapiro, and Vainshtein on the Poisson property of Postnikov's
boundary measurement map. We also show that $B$ and $C$ type networks with
positive weights parametrize the totally nonnegative part of the respective
group. Finally, we construct network parametrizations of double Bruhat cells in
symplectic and odd-dimensional orthogonal groups, and identify the
corresponding face weights with Fock-Goncharov cluster coordinates.
