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Superport networks

Pavlo Pylyavskyy, Svetlana Shirokovskikh, Mikhail Skopenkov

TL;DR

The paper introduces superport networks, a unified framework that extends ordinary and multiport electrical networks by partitioning boundary vertices into superports with zero net current per superport and fixed voltage differences within each superport. It develops the theory of voltages and currents, proving existence/uniqueness and providing forest-based combinatorial formulas for the response, voltages, and currents via Kenyon–Wilson identities. The central contribution is a generalized matrix-tree theorem for superport networks, giving a determinant formula for the response matrix in terms of spanning forests with signed contributions, plus corollaries like a generalized Cayley formula. The work lays groundwork for further study of minors, inverse problems, and network transformations in this broader setting, with potential applications to discrete harmonic functions and multiply-connected domains.

Abstract

We study multiport networks, common in electrical engineering. They have boundary conditions different from electrical networks: the boundary vertices are split into pairs and the sum of the incoming currents is set to be zero in each pair. If one sets the voltage difference for each pair, then the incoming currents are uniquely determined. We generalize Kirchhoff's matrix-tree theorem to this setup. Different forests now contribute with different signs, making the proof subtle. In particular, we use the formula for the response matrix minors by R. Kenyon-D. Wilson, determinantal identities, and combinatorial bijections. We introduce superport networks, generalizing both ordinary networks and multiport ones.

Superport networks

TL;DR

The paper introduces superport networks, a unified framework that extends ordinary and multiport electrical networks by partitioning boundary vertices into superports with zero net current per superport and fixed voltage differences within each superport. It develops the theory of voltages and currents, proving existence/uniqueness and providing forest-based combinatorial formulas for the response, voltages, and currents via Kenyon–Wilson identities. The central contribution is a generalized matrix-tree theorem for superport networks, giving a determinant formula for the response matrix in terms of spanning forests with signed contributions, plus corollaries like a generalized Cayley formula. The work lays groundwork for further study of minors, inverse problems, and network transformations in this broader setting, with potential applications to discrete harmonic functions and multiply-connected domains.

Abstract

We study multiport networks, common in electrical engineering. They have boundary conditions different from electrical networks: the boundary vertices are split into pairs and the sum of the incoming currents is set to be zero in each pair. If one sets the voltage difference for each pair, then the incoming currents are uniquely determined. We generalize Kirchhoff's matrix-tree theorem to this setup. Different forests now contribute with different signs, making the proof subtle. In particular, we use the formula for the response matrix minors by R. Kenyon-D. Wilson, determinantal identities, and combinatorial bijections. We introduce superport networks, generalizing both ordinary networks and multiport ones.
Paper Structure (7 sections, 20 theorems, 35 equations, 10 figures, 1 algorithm)

This paper contains 7 sections, 20 theorems, 35 equations, 10 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. Preliminaries: electrical networks
  3. Quick start: multiport networks
  4. The definition of a superport network
  5. The response of a superport network
  6. Voltages and currents in superport networks
  7. Matrix-tree theorem for superport networks

Key Result

Theorem 2.2

(See, e.g., PS) For any electrical circuit the system of linear equations (C), (I) in the variables $U_k$, where $m+1 \leqslant k\leqslant n$, and $I_{kl}$, where $1\leqslant k,l\leqslant n$, has a unique solution.

Figures (10)

  • Figure 1: Example of a multiport network. The two ports are shown in color. Given the voltage differences $\Delta U_{12}$ and $\Delta U_{34}$, the voltages $U_k$ and the currents $I_{kl}$ for $k,l=1,\dots,6$ are found from the Kirchhoff and Ohm laws and the port condition $I_1=-I_2$, $I_3=-I_4$. See Definition \ref{['def-multiport']}.
  • Figure 2: The Box-H transformation. It preserves the response of the two-port network and also the voltage drop between the ports (shown in color) S.
  • Figure 3: An electrical network. The boundary vertices are 1, 2, 3, 4 (in blue). Examples of valid forests in the network are shown in green.
  • Figure 4: Kirchhoff's matrix-tree theorem. The boundary vertices are 1, 2, 3, 4 (in blue). Altogether, they form the unique valid forest. The spanning trees are in green. See Theorem \ref{['Kirchhoff']}.
  • Figure 5: An electric network (shown in gray) with the boundary vertices decomposed into four sets $X,Y,Z,W$. Here $X=\{x_1,x_2,x_3\}$, $Y=\{y_1,y_2,y_3\}$, $Z=\{z_1,z_2,z_3\}$, and $W=\{w_1,\dots,w_9\}$ are denoted by empty squares, filled squares, empty rhombi, and filled rhombi respectively. The spanning forest shown in black (including isolated vertices) "contributes" to $C_{X, Z}^{Y, Z}$ with the sign $(-1)^{|X|}\mathrm{sgn}(\pi) =-1$. The vertices $x_1,x_2,x_3$ are joined with $y_2,y_3,y_1$ respectively by paths in the forest, hence the permutation $\pi = \left( 123231\right)$, and the forest "contributes" to $w(x_1 y_2| x_2 y_3|x_3 y_1 |w_1|w_2|\dots|w_9)$. See Theorem \ref{['Kenyon-Wilson']}.
  • ...and 5 more figures

Theorems & Definitions (49)

  • Definition 2.1
  • Theorem 2.2
  • Theorem 2.3: Kirchhoff's matrix-tree theorem
  • Corollary 2.4: Cayley's formula
  • proof : Sketch of the proof
  • Theorem 2.5: Kenyon--Wilson's all-minors matrix-tree theorem
  • Remark 2.6
  • Proposition 2.7
  • proof
  • Definition 3.1
  • ...and 39 more