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Are the flows of complex-valued Laplacians and their pseudoinverses related?

Aditi Saxena, Twinkle Tripathy, Rajasekhar Anguluri

Abstract

Laplacian flows model the rate of change of each node's state as being proportional to the difference between its value and that of its neighbors. Typically, these flows capture diffusion or synchronization dynamics and are well-studied. Expanding on these classical flows, we introduce a pseudoinverse Laplacian flow system, substituting the Laplacian with its pseudoinverse within complex-valued networks. Interestingly, for undirected graphs and unsigned weight-balanced digraphs, Laplacian and the pseudoinverse Laplacian flows exhibit an interdependence in terms of consensus. To show this relation, we first present the conditions for achieving consensus in the pseudoinverse Laplacian flow system using the property of real eventually exponentially positivity. Thereafter, we show that the pseudoinverse Laplacian flow system converges to consensus if and only if the Laplacian flow system achieves consensus in the above-mentioned networks. However, these are only the sufficient conditions for digraphs. Further, we illustrate the efficacy of the proposed approach through examples, focusing primarily on power networks.

Are the flows of complex-valued Laplacians and their pseudoinverses related?

Abstract

Laplacian flows model the rate of change of each node's state as being proportional to the difference between its value and that of its neighbors. Typically, these flows capture diffusion or synchronization dynamics and are well-studied. Expanding on these classical flows, we introduce a pseudoinverse Laplacian flow system, substituting the Laplacian with its pseudoinverse within complex-valued networks. Interestingly, for undirected graphs and unsigned weight-balanced digraphs, Laplacian and the pseudoinverse Laplacian flows exhibit an interdependence in terms of consensus. To show this relation, we first present the conditions for achieving consensus in the pseudoinverse Laplacian flow system using the property of real eventually exponentially positivity. Thereafter, we show that the pseudoinverse Laplacian flow system converges to consensus if and only if the Laplacian flow system achieves consensus in the above-mentioned networks. However, these are only the sufficient conditions for digraphs. Further, we illustrate the efficacy of the proposed approach through examples, focusing primarily on power networks.

Paper Structure

This paper contains 10 sections, 9 theorems, 16 equations, 7 figures.

Table of Contents

  1. INTRODUCTION
  2. PRELIMINARIES
  3. Matrix Algebra
  4. Graph Theory
  5. Psuedoinverse Laplacian flows
  6. Pseudoinverses of real eventually exponentially positive Laplacians
  7. Undirected networks
  8. Unsigned directed networks
  9. Simulations
  10. CONCLUSIONS AND FUTURE WORK

Key Result

Theorem 1

Consider $M$ and $M^{H} \in \mathbb{C}^{n \times n}$ such that $\Re(\mathbf{x})\geq\left | \Im(\mathbf{x}) \right |, \Re(\mathbf{z})\geq\left | \Im(\mathbf{z}) \right |$ holds where $\mathbf{x}$ and $\mathbf{z}$ are the dominant right and left eigenvectors of $M$, respectively. Then, the following s where $\mathcal{O}$ is as defined in Eqn.eq:Q.

Figures (7)

  • Figure 1: For the given network, $-L$ is not rEEP, yet consensus is achieved. The trajectories above and below in Fig.\ref{['fig:Lflowcounter']} denote the real and imaginary parts of the states, respectively.
  • Figure 2: Strongly connected weight-balanced digraph
  • Figure 3: Heatmaps showing the matrix structure of $L^\dagger$ for real and imaginary parts respectively in an IEEE 10 bus network (the number here denotes the total nodes)
  • Figure 4: The matrix structure of $\Re(e^{-Lt})$ and $\Re(e^{-L^{\dagger}t}$) at $t=1$ and $t=15$ respectively. Values on the colorbar represent the entire range that encompasses all the values of a matrix, and all of these values are positive implying rEEP property.
  • Figure 5: Consensus achieved by flow systems for the unsigned digraph in Ex. \ref{['ex:unsigned']}. We plot the real and imaginary state trajectories. Observe that the real (upper) and imaginary (lower) states converge to different values.
  • ...and 2 more figures

Theorems & Definitions (20)

  • Definition 1
  • Definition 2
  • Definition 3
  • Theorem 1
  • Lemma 1
  • Lemma 2
  • Lemma 3
  • proof
  • Theorem 2
  • proof
  • ...and 10 more