Internal edge vectors on plabic networks in the disk and a generalization of Talaska formula

Following [42], positroid cells ${\mathcal S}_{\mathcal M}^{\mbox{TNN}}$ in totally non-negative Grassmannians ${Gr^{\mbox{TNN}} (k,n)}$ admit parametrizations by positive weights on planar bicolored directed perfect networks in the disk. An explicit formula for elements of matrices representing the points in ${\mathcal S}_{\mathcal M}^{\mbox{TNN}}$ was obtained in [49] in terms of flows on such networks. The formulas from [42,49] are defined on the boundary edge vectors. In this paper we propose an extension of these formulas for vectors on internal edges defined as summations over paths on the given directed network gauged by the choice of a ray direction. This gauge choice does not affect the boundary edge vectors, which generate the Postnikov boundary measurement map. The systems of internal edge vectors corresponding to different choices of gauge ray directions coincide up to sign, the sign rule admits a simple explicit description. We prove that the components of these edge vectors are rational in the weights with subtraction--free denominators. Moreover, these components are expressed in terms of internal edge flows; these formulas extend the original Talaska ones to the internal edges. These vectors also solve the system of geometric relations associated to the corresponding network. These relations are full rank and respect the total non--negativity property on the full positroid cell. We also provide explicit formulas both for the transformation rules of the edge vectors with respect to the orientation, and for their transformations due to moves and reductions of networks.