Topology Reconstruction of a Resistive Network with Limited Boundary Measurements

We consider the problem of reconstructing all possible topologies of the circular planar passive-resistive network with only
$1\Omega$
resistances, housed inside a black box, with limited boundary measurements. The reconstruction problem is an inverse problem and, in general, has no unique solution. The limitedly available boundary measurements are used to construct a partially known resistance distance matrix. The partially known resistance distance matrix is then related to the unknown Laplacian matrix, resulting in many nonlinear multivariate polynomials. A method is proposed to reconstruct the network topology and edge resistor values simultaneously using the Gröbner basis. Numerical simulation establishes the effectiveness of the proposed strategy.