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On Metzler positive systems on hypergraphs

Shaoxuan Cui, Guofeng Zhang, Hildeberto Jardón-Kojakhmetov, Ming Cao

Abstract

In graph-theoretical terms, an edge in a graph connects two vertices while a hyperedge of a hypergraph connects any more than one vertices. If the hypergraph's hyperedges further connect the same number of vertices, it is said to be uniform. In algebraic graph theory, a graph can be characterized by an adjacency matrix, and similarly, a uniform hypergraph can be characterized by an adjacency tensor. This similarity enables us to extend existing tools of matrix analysis for studying dynamical systems evolving on graphs to the study of a class of polynomial dynamical systems evolving on hypergraphs utilizing the properties of tensors. To be more precise, in this paper, we first extend the concept of a Metzler matrix to a Metzler tensor and then describe some useful properties of such tensors. Next, we focus on positive systems on hypergraphs associated with Metzler tensors. More importantly, we design control laws to stabilize the origin of this class of Metzler positive systems on hypergraphs. In the end, we apply our findings to two classic dynamical systems: a higher-order Lotka-Volterra population dynamics system and a higher-order SIS epidemic dynamic process. The corresponding novel stability results are accompanied by ample numerical examples.

On Metzler positive systems on hypergraphs

Abstract

In graph-theoretical terms, an edge in a graph connects two vertices while a hyperedge of a hypergraph connects any more than one vertices. If the hypergraph's hyperedges further connect the same number of vertices, it is said to be uniform. In algebraic graph theory, a graph can be characterized by an adjacency matrix, and similarly, a uniform hypergraph can be characterized by an adjacency tensor. This similarity enables us to extend existing tools of matrix analysis for studying dynamical systems evolving on graphs to the study of a class of polynomial dynamical systems evolving on hypergraphs utilizing the properties of tensors. To be more precise, in this paper, we first extend the concept of a Metzler matrix to a Metzler tensor and then describe some useful properties of such tensors. Next, we focus on positive systems on hypergraphs associated with Metzler tensors. More importantly, we design control laws to stabilize the origin of this class of Metzler positive systems on hypergraphs. In the end, we apply our findings to two classic dynamical systems: a higher-order Lotka-Volterra population dynamics system and a higher-order SIS epidemic dynamic process. The corresponding novel stability results are accompanied by ample numerical examples.
Paper Structure (15 sections, 25 theorems, 37 equations, 7 figures, 1 table)

This paper contains 15 sections, 25 theorems, 37 equations, 7 figures, 1 table.

Table of Contents

  1. Introduction
  2. Preliminaries on tensors and hypergraphs
  3. Perron– Frobenius Theorem of a Metzler Tensor
  4. Polynomial dynamical systems on a hypergraph
  5. Hypergraph spectrum, strong connectivity, and H-eigenvector centrality
  6. Positive Metzler-tensor-based systems on a uniform hypergraph
  7. Feedback control strategies
  8. Constant control input
  9. On a class of Metzler-tensor-based systems on non-uniform hypergraphs
  10. Application to an SIS epidemic model on a uniform and a non-uniform hypergraph
  11. Application on a cooperative Lotka-Volterra model on a uniform hypergraph
  12. Further discussion
  13. Computation and estimation of H-eigenvalues
  14. Common positive vectors
  15. Conclusions

Key Result

Lemma 1

If $A$ is a nonnegative irreducible tensor of order $m$ dimension $n$ and $\rho(A)$ is its spectral radius, then the following hold:

Figures (7)

  • Figure 1: Example of an 4-uniform, 5-petal(the number of petal) sunflower undirected hypergraph with a singleton core $\{u\}$.
  • Figure 2: Illustration figure of a non-uniform undirected hypergraph. The original hypergraph is illustrated in a). The decomposition into different layers with each layer a uniform hypergraph is illustrated in b). The Hypergraph can be projected as a graph as c).
  • Figure 3: Simulation of system \ref{['eq:sys1']} with $\lambda(A)<0$.
  • Figure 4: Simulation of system \ref{['eq:sys1']} with $\lambda(A)>0$. The time is presented on a logarithmic scale. The divergence rate is faster than exponential.
  • Figure 5: Simulation of system \ref{['eq:affine']} for initial conditions in $\{x|\mathbf{0}<x< x^*\}$ (IC1) and in $\{x|x\geq x^*\}$ (IC2).
  • ...and 2 more figures

Theorems & Definitions (58)

  • Lemma 1: Theorem 3.6 chang2013survey
  • Remark 1
  • Lemma 2: Theorems 2.19 and 2.20 yang2011further
  • Proposition 1
  • proof
  • Theorem 1
  • proof
  • Remark 2
  • Remark 3
  • Lemma 3
  • ...and 48 more