Topology Reconstruction of a Resistor Network with Limited Boundary Measurements: An Optimization Approach

Abstract: A problem of reconstruction of the topology and the respective edge resistance values of an unknown circular planar passive resistive network using limitedly available resistance distance measurements is considered. We develop a multistage topology reconstruction method, assuming that the number of boundary and interior nodes, the maximum and minimum edge conductance, and the Kirchhoff index are known apriori. First, a maximal circular planar electrical network consisting of edges with resistors and switches is constructed; no interior nodes are considered. A sparse difference in convex program $\mathbf{\Pi}_1$ accompanied by round down algorithm is posed to determine the switch positions. The solution gives us a topology that is then utilized to develop a heuristic method to place the interior nodes. The heuristic method consists of reformulating $\mathbf{\Pi}_1$ as a difference of convex program $\mathbf{\Pi}_2$ with relaxed edge weight constraints and the quadratic cost. The interior node placement thus obtained may lead to a non-planar topology. We then use the modified Auslander, Parter, and Goldstein algorithm to obtain a set of planar network topologies and re-optimize the edge weights by solving $\mathbf{\Pi}_3$ for each topology. Optimization problems posed are difference of convex programming problem, as a consequence of constraints triangle inequality and the Kalmansons inequality. A numerical example is used to demonstrate the proposed method.