Generalizations of bounds on the index of convergence to weighted digraphs

Abstract

We study sequences of optimal walks of a growing length, in weighted digraphs, or equivalently, sequences of entries of max-algebraic matrix powers with growing exponents. It is known that these sequences are eventually periodic when the digraphs are strongly connected. The transient of such periodicity depends, in general, both on the size of digraph and on the magnitude of the weights. In this paper, we show that some bounds on the indices of periodicity of (unweighted) digraphs, such as the bounds of Wielandt, Dulmage-Mendelsohn, Schwarz, Kim and Gregory-Kirkland-Pullman, apply to the weights of optimal walks when one of their ends is a critical node.

On the tropical discrete logarithm problem and security of a protocol based on tropical semidirect product

“…The next immediate corollary of above results will be used in practice, for solving the tropical discrete logarithm problem. It is closely related to an observation by Nachtigall [17] that critical rows and columns of matrix powers become periodic after O(d 2 ), and the further more refined results of Merlet et al [14].…”

“…Thus we formulate a tropical discrete logarithm problem and suggest a solution of it based on the weak CSR expansion of Merlet et al [15]. The solution is also very closely related to the quadratic bound on the ultimate periodicity of critical rows and columns of tropical matrix powers obtained by Nachtigall [17] and improved by Merlet et al [14]. Theoretically, the solution is guaranteed to work in some special cases, but it also has 100% success in our numerical experiments.…”

Reachability of eigenspaces for interval circulant matrices in max-algebra

“…when min(a i , a) = a i . Both (12) and (13) are obvious. This shows (11) and hence (10) and (A/λ(A)) ≤ ( Â/λ( Â)).…”

Asymptotic consensus without self-confidence

“…Formally, we denote the smallest T such that there is a walk from i to j of length t for all nodes i and j such that j is reachable from i in G and all t ≥ T by T (G). Wielandt provided an upper bound on the exponent, although many more followed [5,15,10,6,11]. Wielandt's bound is the best possible upper bound in terms of only the number of nodes.…”