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

Shivanagouda Biradar, Deepak U Patil

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 $\mathbfΠ_1$ 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 $\mathbfΠ_1$ as a difference of convex program $\mathbfΠ_2$ 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 $\mathbfΠ_3$ 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.

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 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 as a difference of convex program 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 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.

Paper Structure

This paper contains 30 sections, 8 theorems, 31 equations, 14 figures, 7 algorithms.

Table of Contents

  1. Introduction
  2. Contributions
  3. Mathematical Notations
  4. Problem Formulation
  5. Laplacian and Resistance Distance Matrix
  6. Triangle Inequality & Kalmanson's Inequality
  7. Network Initialization
  8. Construction of $MPRSN$
  9. Component A
  10. Component B
  11. Determining Switch Positions in $\Gamma_M$
  12. Optimization problem $\mathcal{I}$
  13. Optimization problem $\mathbf{\Pi_1}$
  14. Round Down Algorithm
  15. Heuristic of Placement of Interior Nodes
  16. ...and 15 more sections

Key Result

Theorem 2

choi2019resistance For any three distinct boundary nodes$i,j,k$ in CPPR $\Gamma$ such that $1\le i < j < k \le n_b$, the resistance distances $r_{i,j}^d$, $r_{j,k}^d$ and $r_{i,k}^d$ satisfies,

Figures (14)

  • Figure 1: Unknown circular planar graph $\mathcal{G}$.
  • Figure 2: A general construction of resistor switch network.
  • Figure 3: Maximal circular planar graph $\mathcal{G}^{max}_4$.
  • Figure 4: Resistor switch network.
  • Figure 5: $MPRSN$ on $4$ boundary nodes
  • ...and 9 more figures

Theorems & Definitions (13)

  • Theorem 2
  • Theorem 3
  • Definition 4
  • Remark 5
  • Example 6
  • Proposition 7
  • Proposition 8
  • Definition 9: Tree arc & Back edges
  • Proposition 10
  • Proof 1
  • ...and 3 more