On Diagonalizable Systems with Random Structure

TL;DR

This paper investigates the likelihood of structural diagonalizability in systems modeled by Erdős-Rényi graphs, revealing that dense graphs are almost always diagonalizable, while sparse ones are not, supported by theoretical analysis and simulations.

Contribution

It provides a comprehensive probabilistic analysis of structural diagonalizability in random graph-based systems, extending graph-theoretic characterizations to different edge-density regimes.

Findings

Dense graphs are almost always structurally diagonalizable.

Probability of diagonalizability decreases with graph sparsity.

Theoretical results are validated through extensive simulations.

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 Erd\H{o}s-R\'{e}nyi 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.