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On Diagonalizable Systems with Random Structure

Yuan Zhang, Yutong Han, Yuanqing Xia, Aming Li

TL;DR

This work analyzes the probability that structured linear systems, represented by directed Erdős–Rényi graphs, are structurally diagonalizable. By linking diagonalizability to Hamiltonian decompositions and consistent matchings in a bipartite representation, it derives precise asymptotics across dense, medium, and sparse regimes for both $ ext{G}(n,p)$ and $ ext{G}(n,p,q)$. The results show near-certain diagonalizability in dense graphs, tight bounds in medium-density regimes, and vanishing probability in sparse graphs, with simulations validating the theory. Practically, the findings imply polynomial-time verification for SOC and SFO properties in almost all large random networks, and they resolve conjectures about diagonalizability in random directed graphs.

Abstract

Diagonalizability plays an important role in the analysis and design of multivariable systems. A structured matrix is called structurally diagonalizable if almost all of its numerical realizations, obtained by assigning real values to its free entries, are diagonalizable. Structural diagonalizability is useful for the verification and optimization of various structural system properties. In this paper, we study the asymptotic probability distribution of structural diagonalizability for structured systems whose system matrices are represented by directed Erdős-Rényi random graphs. Leveraging a recently established graph-theoretic characterization of structural diagonalizability, we analyze the distribution of structurally diagonalizable graphs under different edge-density regimes. For dense graphs, we prove that the system is almost always structurally diagonalizable. For graphs of medium density, we derive tight upper and lower bounds on the asymptotic probability of structural diagonalizability. For extremely sparse graphs, we show that this probability approaches 0. The theoretical results are validated through extensive numerical simulations with varying numbers of vertices and connection probabilities.

On Diagonalizable Systems with Random Structure

TL;DR

This work analyzes the probability that structured linear systems, represented by directed Erdős–Rényi graphs, are structurally diagonalizable. By linking diagonalizability to Hamiltonian decompositions and consistent matchings in a bipartite representation, it derives precise asymptotics across dense, medium, and sparse regimes for both and . The results show near-certain diagonalizability in dense graphs, tight bounds in medium-density regimes, and vanishing probability in sparse graphs, with simulations validating the theory. Practically, the findings imply polynomial-time verification for SOC and SFO properties in almost all large random networks, and they resolve conjectures about diagonalizability in random directed graphs.

Abstract

Diagonalizability plays an important role in the analysis and design of multivariable systems. A structured matrix is called structurally diagonalizable if almost all of its numerical realizations, obtained by assigning real values to its free entries, are diagonalizable. Structural diagonalizability is useful for the verification and optimization of various structural system properties. In this paper, we study the asymptotic probability distribution of structural diagonalizability for structured systems whose system matrices are represented by directed Erdős-Rényi random graphs. Leveraging a recently established graph-theoretic characterization of structural diagonalizability, we analyze the distribution of structurally diagonalizable graphs under different edge-density regimes. For dense graphs, we prove that the system is almost always structurally diagonalizable. For graphs of medium density, we derive tight upper and lower bounds on the asymptotic probability of structural diagonalizability. For extremely sparse graphs, we show that this probability approaches 0. The theoretical results are validated through extensive numerical simulations with varying numbers of vertices and connection probabilities.
Paper Structure (18 sections, 20 theorems, 68 equations, 10 figures, 1 table)

This paper contains 18 sections, 20 theorems, 68 equations, 10 figures, 1 table.

Table of Contents

  1. Introduction
  2. Preliminaries: Structurally diagonalizable directed graphs
  3. Random graphs
  4. The $\mathcal{G}(n, p, q)$ random graph model
  5. The $\mathcal{G}(n, p)$ random graph model
  6. Graph-theoretic criteria for structural diagonalizability
  7. Asymptotic analysis of structural diagonalizability probability
  8. Structural diagonalizability analysis in $\mathcal{G}(n,p)$
  9. Structural diagonalizability analysis of dense graphs
  10. Structural diagonalizability analysis of medium graphs
  11. Structural diagonalizability analysis of sparse graphs
  12. Structural diagonalizability analysis in $\mathcal{G}(n,p,q)$
  13. Experiments and validation
  14. The simulation results of the $\mathcal{G}(n,p)$) random graph model
  15. The simulation results of the $\mathcal{G}(n,p,q)$ random graph model
  16. ...and 3 more sections

Key Result

Lemma 2.3

A structured matrix $\bar{A} \in \{0, \ast\}^{n \times n}$ is structurally diagonalizable if and only if the following equivalent conditions hold:

Figures (10)

  • Figure 1: Directed graph representations of the structured matrices $\bar{A}_{\alpha}$ and $\bar{A}_{\beta}$
  • Figure 1: The case of digraphs that are not structurally diagonalizable, as proposed in \ref{['thm:shangjie']}.
  • Figure 1: Comparison between simulation results and theoretical upper/lower bounds for $p=\frac{\log n+c+o(1)}{n}$ with fixed $n=2000$ and varying $c$
  • Figure 2: Consistent and non-consistent matchings in a bipartite graph: (a) size-$3$ consistent matching, (b) size-$3$ non-consistent matching.
  • Figure 2: Comparison between simulation results and theoretical results for $p=\frac{c}{n}$ with fixed $n=2000$ and varying $c$
  • ...and 5 more figures

Theorems & Definitions (40)

  • Definition 2.1
  • Definition 2.2
  • Lemma 2.3: taken from zhang2025generic
  • Lemma 4.1: Sufficient condition for structural diagonalizability
  • Proof 1
  • Lemma 4.2: Necessary and sufficient condition for the existence of a Hamiltonian decomposition, reinschke1988multivariable
  • Lemma 4.3: Equivalent conditions for the nonexistence of a Hamiltonian decomposition
  • Proof 2
  • Definition 4.4
  • Lemma 4.5
  • ...and 30 more