Topology reconstruction of a resistor network with limited boundary measurements: An optimization approach

We address the problem of reconstructing the topology and edge resistance values of an unknown CPPR network using limited resistance distance measurements. A multistage topology reconstruction method is developed, assuming prior knowledge of the number of boundary and interior nodes, the maximum and minimum edge conductance, and the Kirchhoff index. The process begins by constructing a maximal circular planar network with edges composed of resistors and switches, excluding interior nodes. A sparse difference of convex programming problem Π 1 combined with a round-down algorithm is formulated to determine optimal switch positions, yielding an initial topology. This topology serves as the basis for a heuristic method to place interior nodes. The heuristic involves reformulating Π 1 into a new optimization problem Π 2 with relaxed edge weight constraints and a quadratic cost function. The resulting placement of interior nodes may produce a non-planar topology. To address this, we develop a modified Auslander, Parter, and Goldstein algorithm to generate a set of planar network topologies. For each planar topology, edge weights are re-optimized by solving a third optimization problem Π 3 . These optimization problems are formulated as difference of convex programming problems, incorporating triangle inequality and Kalmanson’s inequality constraints. A numerical example demonstrates the effectiveness of the proposed method.