On the Transience of Linear Max-Plus Dynamical Systems

We study the transients of linear max-plus dynamical systems. For that, we consider for each irreducible max-plus matrix A, the weighted graph G(A) such that A is the adjacency matrix of G(A). Based on a novel graph-theoretic counterpart to the number-theoretic Brauer’s theorem, we propose two new methods for the construction of arbitrarily long paths in G(A) with maximal weight. That leads to two new upper bounds on the transient of a linear max-plus system which both improve on the bounds previously given by Even and Rajsbaum (STOC 1990, Theory of Computing Systems 1997), by Bouillard and Gaujal (Research Report 2000), and by Soto y Koelemeijer (PhD Thesis 2003), and are, in general, incomparable with Hartmann and Arguelles’ bound (Mathematics of Operations Research 1999). With our approach, we also show how to improve the latter bound by a factor of two. A significant benefit of our bounds is that each of them turns out to be linear in the size of the system in various classes of linear max-plus system whereas the bounds previously given are all at least quadratic. Our second result concerns the relationship between matrix and system transients: We prove that the transient of an NxN matrix A is, up to some constant, equal to the transient of an A-linear system with an initial vector whose norm is quadratic in N. Finally, we study the applicability of our results to the well-known Full Reversal algorithm whose behavior can be described as a min-plus linear system.

💡 Research Summary

The paper investigates the transient phase of linear max‑plus dynamical systems by translating the problem into a graph‑theoretic setting. For any irreducible max‑plus matrix A, the associated weighted digraph G(A) is considered, and its critical subgraph G_c (the subgraph formed by edges belonging to cycles with maximal average weight) is identified as the key structure governing long‑term periodicity.

The authors introduce two novel techniques for constructing arbitrarily long maximum‑weight paths in G(A). Both techniques rely on a graph‑theoretic analogue of Brauer’s number‑theoretic theorem, which yields a new parameter called the “exploration penalty” k. This k is the smallest integer such that for every node i and every length n > k that is a multiple of the graph’s cyclicity, there exists a closed walk of length n starting and ending at i. By applying a generalized Brauer lemma, the authors bound k by O(|V|²), dramatically improving over earlier quadratic or exponential bounds.

1. Explorative Method – The first method systematically visits all strongly connected components of the critical subgraph. It uses the girth, circumference, and diameter of each component to guarantee the existence of long optimal walks that repeatedly enter and exit the critical region. This yields an upper bound on the transient that is linear in the number of nodes for several important families of graphs (e.g., trees, graphs with bounded girth).

2. Repetitive Method – The second method confines the walk to a single elementary critical cycle, repeating it as many times as needed. By carefully aligning the cycle length with the overall cyclicity, the authors improve the classic Hartmann‑Arguelles bound by a factor of two. Although this bound is incomparable with the explorative bound in the general case, it becomes optimal for primitive graphs where the cyclicity equals one.

Beyond these two bounds, the paper establishes a quantitative relationship between matrix transients and system transients. It proves that for an N × N irreducible matrix A, the transient of A is, up to a constant factor, equal to the transient of an A‑linear system whose initial vector has norm Θ(N²). Consequently, any upper bound derived for system transients can be transferred to matrix transients without resorting to the binary‑search algorithms previously used by Hartmann and Argues.

The theoretical developments are illustrated on the Full Reversal algorithm, a well‑known distributed protocol whose dynamics can be modeled as a min‑plus linear system. By applying the new transient bounds, the authors obtain tight estimates for the termination time of Full Reversal in both routing and scheduling contexts, showing that in many practical topologies the termination time grows linearly with the number of nodes.

In summary, the paper makes three major contributions: (i) a graph‑theoretic extension of Brauer’s theorem leading to the exploration‑penalty concept; (ii) two distinct upper bounds on the transient of max‑plus systems that improve upon all previously known bounds and are linear for broad graph classes; and (iii) a clear bridge between matrix and system transients, together with an application to a concrete distributed algorithm. These results advance both the theoretical understanding of max‑plus algebra and its practical analysis in distributed computing and networked control systems.