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On an inverse tridiagonal eigenvalue problem and its application to synchronization of network motion

Luca Dieci, Cinzia Elia, Alessandro Pugliese

TL;DR

This work addresses an inverse eigenvalue problem for unreduced tridiagonal matrices to realize a Laplacian $L$ with $0=\lambda_1<\lambda_2<\cdots<\lambda_N$ and $L\mathbf{e}=0$, then uses the resulting $L$ to analyze synchronization stability via the Master Stability Function (MSF) in networks with nearest-neighbor coupling. It introduces two constructive algorithms: diag2trid to obtain symmetric unreduced tridiagonal matrices with a prescribed spectrum, and TridZeroRowSum to obtain a (generally non-symmetric) tridiagonal with a specified null-vector, while examining the feasibility of symmetry. The paper proves a fundamental bound showing that among symmetric tridiagonal network Laplacians the eigenvalue ratio is maximized by the standard diffusive matrix, implying symmetry can prevent achieving a negative MSF for some spectra. Numerical experiments on Van der Pol and Rössler networks demonstrate that non-symmetric tridiagonal $L$ can yield negative MSF intervals and achieve synchronization with modest coupling, whereas symmetric cases may require large coupling or fail to synchronize. Overall, the work connects inverse-EBP constructions to network synchronization, offering practical guidance on choosing tridiagonal weights to realize stable synchronous motion and highlighting symmetry-induced limitations for real-world networks.

Abstract

In this work, motivated by the study of stability of the synchronous orbit of a network with tridiagonal Laplacian matrix, we first solve an inverse eigenvalue problem which builds a tridiagonal Laplacian matrix with eigenvalues $λ_1=0<λ_2<\cdots <λ_N$ and null-vector $\boldsymbol{e} = \begin{bmatrix} 1 \\ \vdots \\ 1 \end{bmatrix}$. Then, we show how this result can be used to guarantee -- if possible -- that a synchronous orbit of a connected tridiagonal network associated to the matrix $L$ above is asymptotically stable, in the sense of having an associated negative Master Stability Function (MSF). We further show that there are limitations when we also impose symmetry for $L$.

On an inverse tridiagonal eigenvalue problem and its application to synchronization of network motion

TL;DR

This work addresses an inverse eigenvalue problem for unreduced tridiagonal matrices to realize a Laplacian with and , then uses the resulting to analyze synchronization stability via the Master Stability Function (MSF) in networks with nearest-neighbor coupling. It introduces two constructive algorithms: diag2trid to obtain symmetric unreduced tridiagonal matrices with a prescribed spectrum, and TridZeroRowSum to obtain a (generally non-symmetric) tridiagonal with a specified null-vector, while examining the feasibility of symmetry. The paper proves a fundamental bound showing that among symmetric tridiagonal network Laplacians the eigenvalue ratio is maximized by the standard diffusive matrix, implying symmetry can prevent achieving a negative MSF for some spectra. Numerical experiments on Van der Pol and Rössler networks demonstrate that non-symmetric tridiagonal can yield negative MSF intervals and achieve synchronization with modest coupling, whereas symmetric cases may require large coupling or fail to synchronize. Overall, the work connects inverse-EBP constructions to network synchronization, offering practical guidance on choosing tridiagonal weights to realize stable synchronous motion and highlighting symmetry-induced limitations for real-world networks.

Abstract

In this work, motivated by the study of stability of the synchronous orbit of a network with tridiagonal Laplacian matrix, we first solve an inverse eigenvalue problem which builds a tridiagonal Laplacian matrix with eigenvalues and null-vector . Then, we show how this result can be used to guarantee -- if possible -- that a synchronous orbit of a connected tridiagonal network associated to the matrix above is asymptotically stable, in the sense of having an associated negative Master Stability Function (MSF). We further show that there are limitations when we also impose symmetry for .
Paper Structure (12 sections, 10 theorems, 52 equations, 2 figures, 2 algorithms)

This paper contains 12 sections, 10 theorems, 52 equations, 2 figures, 2 algorithms.

Table of Contents

  1. A linear algebra and a dynamical systems problem
  2. Linear algebra background
  3. Networks and the MSF
  4. Our inverse network tridiagonal eigenvalue problem
  5. Symmetric unreduced tridiagonal with given spectrum
  6. Singular tridiagonal with specific null-vector
  7. Impact of symmetry
  8. Symmetric Tridiagonal Networks and the MSF
  9. Numerical Results
  10. Van der Pol
  11. Rössler
  12. Conclusions

Key Result

Lemma 1.4

Given a general tridiagonal, unreduced, matrix $L$ as in GivenTrid, then its eigenvalues are real and distinct, hence $L$ is diagonalizable by a real matrix of eigenvectors $V$: $V^{-1}LV=\operatorname{diag}(\lambda_i, \ i=1,\dots, N)$. Moreover, the eigenvalues do not change as long as the products

Figures (2)

  • Figure 1: Van der Pol, N=32. Left: MSF. Right: 2-norm of the difference between agents.
  • Figure 2: Left: MSF for Rössler. Right: 2-Norm of the difference between agents.

Theorems & Definitions (27)

  • Definition 1.1
  • Definition 1.2
  • Remark 1.3
  • Lemma 1.4
  • proof
  • Example 1.5
  • Definition 1.6
  • Remark 1.7
  • Lemma 2.2
  • proof
  • ...and 17 more