On Diagonalizable Systems with Random Structure

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.

💡 Research Summary

The paper investigates the probability that a structured matrix, whose zero‑nonzero pattern is dictated by a random directed graph, is structurally diagonalizable—that is, almost all numerical realizations of the matrix are diagonalizable. Structural diagonalizability is a generic property: either almost every realization is diagonalizable or almost none are. The authors exploit a recent graph‑theoretic characterization (Lemma 2.3) which states that a structured matrix is structurally diagonalizable if and only if (i) its generic rank equals the maximum number of vertices that can be covered by a collection of vertex‑disjoint directed cycles in the associated digraph, and (ii) the bipartite representation of the digraph admits a maximum matching that is “consistent,” i.e., it corresponds to a set of disjoint cycles covering exactly the matched vertices.

The study focuses on two random graph models: G(n, p, q), where each possible non‑self‑loop edge appears independently with probability p and each self‑loop with probability q, and the simpler G(n, p) (with q = 0). By varying the edge‑density parameter p as a function of the number of vertices n, three regimes are identified:

1. Dense regime (p ≫ log n / n). In this region the expected degree grows faster than log n, guaranteeing with high probability that Hall’s condition for a perfect matching holds in the bipartite graph B(G). Consequently a consistent perfect matching exists, and the digraph can be decomposed into vertex‑disjoint cycles covering all vertices. The generic rank condition is also satisfied, leading to structural diagonalizability with probability tending to 1 as n → ∞.

2. Sparse regime (p ≪ 1 / n). Here the average degree is below one, so the random digraph consists mainly of trees and isolated vertices. The probability of forming any directed cycle vanishes, and the maximum matching size is far below n. Both graph‑theoretic conditions fail with overwhelming probability, and the structural diagonalizability probability converges to 0.

3. Intermediate regime (log n / n ≪ p ≪ 1). This is the most delicate case. The authors employ probabilistic combinatorial tools to bound the likelihood that the random digraph contains a sufficient number of disjoint cycles and that a large consistent matching exists. They introduce the notion of a “consistent‑matching ratio” and derive explicit upper and lower bounds on the asymptotic probability Pₙ(p). Roughly, when p = c·log n / n with c > 1, the probability of structural diagonalizability approaches 1 − exp(−Θ(c log n)), whereas for c < 1 it decays like exp(−Θ((1−c) log n)). Thus a sharp phase transition occurs around the critical threshold p ≈ log n / n.

To validate the theoretical findings, extensive Monte‑Carlo simulations are performed for n = 50, 100, 200 and a wide range of p values, each repeated 10⁴ times. The empirical frequencies of structural diagonalizability match the derived bounds closely, confirming the phase‑transition behavior. Additional experiments vary the self‑loop probability q, showing that larger q slightly raises the probability by facilitating cycle formation, but does not alter the qualitative regime boundaries.

The paper’s contributions are threefold: (i) it translates the algebraic notion of structural diagonalizability into concrete graph‑theoretic criteria involving matchings and cycle covers; (ii) it provides a rigorous probabilistic analysis of these criteria for Erdős–Rényi directed graphs, yielding exact asymptotic probabilities in dense and sparse limits and tight bounds in the intermediate regime; (iii) it demonstrates through simulations that the theoretical predictions are accurate for practical network sizes.

These results have practical implications for the design and analysis of large‑scale networked control systems, sensor networks, and other multi‑agent platforms where the interconnection topology may be random or subject to failures. Knowing that a dense random topology almost surely guarantees structural diagonalizability allows engineers to rely on modal decoupling, simplified controller synthesis, and polynomial‑time verification of output controllability or functional observability. Conversely, in extremely sparse networks, alternative strategies must be considered because diagonalization‑based techniques are unlikely to be applicable. The intermediate regime analysis offers quantitative guidance on how dense the connectivity must be to achieve a desired confidence level in structural diagonalizability.

Overall, the work bridges structured systems theory and random graph theory, opening a new avenue for probabilistic assessment of fundamental algebraic properties in stochastic networked systems.