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Boundary measurements of positive networks on a cylinder of maximal rank 2 and 3

David Whiting

TL;DR

Let $N$ be a network embedded on a cylinder with directed edges and boundary sources and sinks on opposite circles; its boundary measurement matrix $M$ encodes weighted path sums. The authors develop a constructive, column-operator framework that reduces any boundary measurement matrix to an identity matrix using four elementary column operations, while tracking the sign-structure via the cyclic sign variation $\mathrm{cvar}$ and the signs of odd-dimensional minors. The main results are: (i) for $m=2$, $M$ is a boundary measurement matrix iff $\mathrm{cvar}(M)=2$, and (ii) for $m=3$, $M$ is a boundary measurement matrix iff every odd-dimensional minor is nonnegative, connecting boundary measurements on cylinders to totally nonnegative Grassmannians and Poisson/cluster structures. This work extends Postnikov’s disk results to annuli, elucidating the relationship between boundary measurements, column-operations reductions, and the underlying algebraic-geometric framework.

Abstract

Boundary measurement matrices associated to networks on a plane correspond to certain totally nonnegative Grassmannians as shown previously by A. Postnikov. In this paper, we look to generalize this result by categorizing the boundary measurements associated to networks on a cylinder of maximal rank 2 and 3. In particular, we show that the maximal rank 3 matrices associated to networks on a cylinder are precisely the matrices in which every odd-dimensional minor is nonnegative.

Boundary measurements of positive networks on a cylinder of maximal rank 2 and 3

TL;DR

Let be a network embedded on a cylinder with directed edges and boundary sources and sinks on opposite circles; its boundary measurement matrix encodes weighted path sums. The authors develop a constructive, column-operator framework that reduces any boundary measurement matrix to an identity matrix using four elementary column operations, while tracking the sign-structure via the cyclic sign variation and the signs of odd-dimensional minors. The main results are: (i) for , is a boundary measurement matrix iff , and (ii) for , is a boundary measurement matrix iff every odd-dimensional minor is nonnegative, connecting boundary measurements on cylinders to totally nonnegative Grassmannians and Poisson/cluster structures. This work extends Postnikov’s disk results to annuli, elucidating the relationship between boundary measurements, column-operations reductions, and the underlying algebraic-geometric framework.

Abstract

Boundary measurement matrices associated to networks on a plane correspond to certain totally nonnegative Grassmannians as shown previously by A. Postnikov. In this paper, we look to generalize this result by categorizing the boundary measurements associated to networks on a cylinder of maximal rank 2 and 3. In particular, we show that the maximal rank 3 matrices associated to networks on a cylinder are precisely the matrices in which every odd-dimensional minor is nonnegative.

Paper Structure

This paper contains 5 sections, 15 theorems, 21 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Column Operations
  4. 2xn Case
  5. 3xn Case

Key Result

Theorem 1.1

Let $M$ be an $m \times n$ real-valued matrix of maximal rank $m \in \{2,3\}$.

Figures (3)

  • Figure 1: A network in a cylinder together with its boundary measurement matrix $B$. The top and bottom dotted edges of the rectangle are glued together to create the cylinder.
  • Figure 2: A network $N$ with boundary measurement matrix $M$ (left) and the network $N_t$ with boundary measurement matrix $M_t$ (right).
  • Figure 3: An illustration of the column operations used to construct a network $N$.

Theorems & Definitions (46)

  • Theorem 1.1
  • Definition 2.1
  • Definition 2.2
  • Remark 2.3
  • Definition 2.4
  • Definition 2.5
  • Definition 2.6
  • Lemma 2.7
  • Remark 2.8
  • Definition 3.1
  • ...and 36 more