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arxiv: 2412.02315 · v2 · submitted 2024-12-03 · 📡 eess.SY · cs.SY

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

Shivanagouda Biradar , Deepak U Patil This is my paper

Pith reviewed 2026-05-23 08:12 UTC · model grok-4.3

classification 📡 eess.SY cs.SY
keywords resistor networktopology reconstructiondifference of convex programmingboundary measurementsplanar networksKirchhoff indexoptimizationelectrical networks
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The pith

A multistage optimization approach reconstructs the topology of a circular planar resistor network from limited boundary measurements.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper develops a method to reconstruct both the connections and the resistance values in an unknown circular planar resistive network when only some resistance distance measurements from boundary nodes are available. It assumes prior knowledge of node counts, conductance bounds, and the Kirchhoff index to set up a sequence of difference-of-convex optimization problems. The approach first builds a maximal network without interior nodes and solves for edge presence using a sparse DC program, then heuristically places interior nodes, enforces planarity via a modified algorithm, and re-optimizes weights. This allows recovery of the network structure under those assumptions using only boundary data.

Core claim

The central claim is that the topology and edge resistances of an unknown circular planar passive resistive network can be reconstructed using a multistage method that poses difference of convex programs Π1, Π2, and Π3, combined with a heuristic for interior node placement and the modified Auslander-Parter-Goldstein algorithm for planarity, given known numbers of nodes, conductance bounds, and Kirchhoff index.

What carries the argument

Difference-of-convex programming problems (Π1 for switch positions, Π2 for node placement, Π3 for weight re-optimization) applied to a maximal circular planar network with switches.

If this is right

  • The network topology can be recovered even with limited boundary measurements if the priors hold.
  • Interior nodes can be placed heuristically by relaxing constraints in a DC program.
  • Multiple planar topologies can be generated and evaluated by solving Π3 for each.
  • The method works for passive resistive networks that are circular planar.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • This could extend to networks where some priors are estimated rather than known exactly.
  • The approach might apply to non-planar or 3D resistor networks with modifications to the planarity step.
  • Numerical stability of the DC programs could be tested on larger networks.

Load-bearing premise

The number of boundary and interior nodes, the maximum and minimum edge conductances, and the Kirchhoff index must be known in advance.

What would settle it

A counterexample network where the method, given correct priors, outputs a different topology or resistance values than the true one when tested on simulated boundary measurements.

Figures

Figures reproduced from arXiv: 2412.02315 by Deepak U Patil, Shivanagouda Biradar.

Figure 1
Figure 1. Figure 1: Unknown circular planar graph G. A be the set of boundary nodes available for voltage and current measurements. Then, the set of all nodes that are not available is U = V ∖ A. Denote by UB ⊆ U the set of boundary nodes not available for the measurements. The conductivity function γ ∶ E → R+ , assigns to each edge σ ∈ E, a positive real number γ(σ), known as the conductance of σ. The resistance of the edge … view at source ↗
Figure 2
Figure 2. Figure 2: A general construction of resistor switch network. [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Maximal circular planar graph Gmax 4 . The second step is constructing the resistor switch network Cij across the boundary nodes i, j ∈ VB, based on rmax, as shown in Fig.4. Component A of Cij has three switches l ipj 1 ,lipj 2 ,lipj 3 , therefore, there are 2 3 switch combinations that induce 2 3 resistance values. These resistance val￾ues are {∞, 1, 2, 0.666, 3, 0.75, 1.66, 0.625}. The minimum value 0.62… view at source ↗
Figure 4
Figure 4. Figure 4: Resistor switch network [PITH_FULL_IMAGE:figures/full_fig_p006_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: MP RSN on 4 boundary nodes Let the constructed MP RSN be called as ΓM = (GM, γM). Where GM = (VM, EM), VM is the set of all nodes and EM is a set of all edge in ΓM. Let SM ⊂ EM is a set of all node pairs connected through a resistor and switch, for example, in Fig.4, a node pair i and p j 1 are connected by a 1Ω resistor and a switch, therefore ipj 1 ∈ SM. The conductance function γM ∶ C → R+ . The Laplaci… view at source ↗
Figure 6
Figure 6. Figure 6: Topology Reconstruction Example [PITH_FULL_IMAGE:figures/full_fig_p009_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Palm tree representation P of Gˆaux. Bold edges are the tree arcs and dashed edges are the back edges. 6. Rewiring For every graph Gˆp aux,i ∈ Gˆp aux, where Gˆp aux,i = (VB ⋃VI, Eˆp aux,i), construct a resistor network Γˆ i = (Gˆp aux,i, γˆi), where ˆγi ∶ Eˆp aux,i → R+ ,∀1 ≤ i ≤ ∣Gˆp aux∣. The conductivity function ˆγi is unknown. Therefore, let cˆi be a vector of unknown conductances of i th network Γˆ … view at source ↗
Figure 8
Figure 8. Figure 8: Topology Reconstruction Example. valid network, we introduce the triangle and Kalmansons inequality constraints. 8.2. Methodology The methodology proposed works well for small CP P R networks. As the number of boundary nodes increases, the number of edges in the maximal planar graph increases al￾most exponentially. The lower bound on the number of edges for a given number of vertices is given in [35]. In t… view at source ↗
Figure 9
Figure 9. Figure 9: Value of f0 (corresponding to Γ⋆ ) w.r.t number of resis￾tances in component B Initial guess values fed into the optimization routine must be chosen judiciously, for proper network reconstruc￾tion. The quality of reconstructed network Γ⋆ is sensitive to the choice of guesstimate of switch variables vector ρ for Π1. Therefore, a novel method for choosing a proper initial guess of switch positions in ΓM for … view at source ↗
read the original 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.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 1 minor

Summary. The paper claims to develop a multistage optimization method for reconstructing the topology and edge resistances of a circular planar resistive network from limited boundary resistance-distance measurements. It assumes a priori knowledge of the numbers of boundary and interior nodes, bounds on edge conductances, and the Kirchhoff index. The pipeline solves a difference-of-convex program Π1 (with rounding) on a maximal network to select edges, applies a heuristic Π2 to place interior nodes, uses a modified Auslander-Parter-Goldstein procedure to enforce planarity, and re-optimizes conductances via Π3; the approach is illustrated by a numerical example.

Significance. If the a priori quantities can be independently supplied and the DC programs reliably recover the correct topology, the method would offer a concrete optimization-based procedure for a class of network-identification problems. The explicit use of triangle and Kalmanson inequalities to obtain DC formulations is a technical feature that could be reusable, but the dependence on global scalars not derivable from the stated limited measurements restricts the result to a narrow setting.

major comments (2)
  1. [Abstract] Abstract: the reconstruction pipeline is conditioned on knowing the Kirchhoff index a priori, yet the only data source is a proper subset of boundary-to-boundary resistance distances. No procedure is supplied for recovering this global scalar from the given measurements, rendering the input requirements inconsistent with the problem formulation.
  2. [Abstract] Abstract (numerical example): the demonstration is described only at the level of 'a numerical example is used'; no error metrics, recovery rates, or verification that the recovered topology matches the ground truth under the stated assumptions are provided, leaving the empirical support for the central claim unassessed.
minor comments (1)
  1. The modifications made to the Auslander-Parter-Goldstein algorithm are not detailed; a brief description of the changes would improve reproducibility.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the detailed review and constructive comments. We respond to each major comment below, maintaining consistency with the manuscript's stated assumptions and content.

read point-by-point responses

  1. Referee: [Abstract] Abstract: the reconstruction pipeline is conditioned on knowing the Kirchhoff index a priori, yet the only data source is a proper subset of boundary-to-boundary resistance distances. No procedure is supplied for recovering this global scalar from the given measurements, rendering the input requirements inconsistent with the problem formulation.

    Authors: The manuscript explicitly formulates the problem under the assumption that the Kirchhoff index (along with node counts and conductance bounds) is known a priori. The reconstruction method is developed and analyzed within this setting, using the limited boundary measurements as the primary data source. This is consistent with the problem statement, which does not claim that all required scalars must be derived solely from the measurements. The approach applies when these quantities can be supplied independently, as is common in certain network identification scenarios. No inconsistency exists with the stated formulation. revision: no

  2. Referee: [Abstract] Abstract (numerical example): the demonstration is described only at the level of 'a numerical example is used'; no error metrics, recovery rates, or verification that the recovered topology matches the ground truth under the stated assumptions are provided, leaving the empirical support for the central claim unassessed.

    Authors: The abstract is a concise summary and refers to the detailed numerical example in Section 5 of the manuscript, which includes verification that the recovered topology matches the ground truth under the assumptions. To enhance the abstract's clarity regarding empirical support, we will revise it to briefly note the verification and key outcomes from the example. revision: yes

Circularity Check

0 steps flagged

No significant circularity; assumptions stated explicitly as inputs

full rationale

The paper states its multistage optimization method (Π1, Π2, Π3) under explicit a priori assumptions on node counts, conductance bounds, and Kirchhoff index. These are presented as known inputs rather than quantities derived or fitted within the derivation chain. No step reduces a claimed prediction or result to the measurements by construction, no self-citation load-bearing is present, and no ansatz or uniqueness theorem is smuggled in. The derivation is self-contained given the stated assumptions; any concern that the assumptions cannot be obtained from limited measurements is a question of problem formulation or correctness, not circularity in the mathematical steps.

Axiom & Free-Parameter Ledger

3 free parameters · 2 axioms · 0 invented entities

The reconstruction depends on several a priori known quantities that are not derived within the method and on domain assumptions about network geometry and passivity.

free parameters (3)
  • number of boundary and interior nodes
    Assumed known a priori as input to the method
  • maximum and minimum edge conductance
    Assumed known a priori to constrain the optimization
  • Kirchhoff index
    Assumed known a priori and used in the reconstruction
axioms (2)
  • domain assumption The unknown network is circular planar and passive resistive
    Stated as the problem setting in the abstract
  • domain assumption Limited resistance distance measurements are available only at boundary nodes
    Core input assumption for the reconstruction

pith-pipeline@v0.9.0 · 5777 in / 1425 out tokens · 55026 ms · 2026-05-23T08:12:52.575437+00:00 · methodology

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    the edge st can be deleted, or, 2. edge resistance r(n)(st) is increased by 1Ω. Let us call an operation in point 1 as OP1 and, operation in point 2 as OP2. In case- 1, first implement operation OP1, then operation OP2 on Γ (n) I = ( G(n) I , γ(n)) independently. Let ˜rd s,t(OP1) and ˜rd s,t(OP2) be the resistance distance error across s, t∈VB after commi...

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