  Postnikov's plabic graphs in a disk are used to parametrize totally positive
Grassmannians. In recent years plabic graphs have found numerous applications
in math and physics. One of the key features of the theory is the fact that if
a plabic graph is reduced, the face weights can be uniquely recovered from
boundary measurements. On surfaces more complicated than a disk this property
is lost. In this paper we undertake a comprehensive study of a certain
semi-local transformation of weights for plabic networks on a cylinder that
preserve boundary measurements. We call this a plabic R-matrix. We show that
plabic R-matrices have underlying cluster algebra structure, generalizing
recent work of Inoue-Lam-Pylyavskyy. Special cases of transformations we
consider include geometric R-matrices appearing in Berenstein-Kazhdan theory of
geometric crystals, and also certain transformations appearing in a recent work
of Goncharov-Shen.
