Max-algebra and pairwise comparison matrices, II

Combinatorial types of tropical eigenvectors

“…In this formula, φ ii * denotes the directed path from i to i * in the graph of φ, and B = (b ij ) = (a ij − λ φ (A)), where λ φ (A) is the mean of the cycle in the graph of φ with edge weights (a ij ). The inequality representation (7) will imply that the cone Ω φ is linearly isomorphic to R n × R n(n−1) 0 because there are n(n − 1) non-edges (i, j) ∈ [n] 2 \φ.…”

“…Without loss of generality, let (1 → 2 → · · · → k → 1) be the cycle in φ, and (i 1 → i 2 → · · · → i m → i 1 ) the cycle in ψ. Assuming they are different, the inequality in (7) holds strictly for at least one edge in ψ. Using the identities…”

“…, 0] T . The condition (8) is equivalent to that in (7), with strict inequalities for {(i, j) : j = φ(i)}, and it holds if and only if C is in the interior of Ω φ .…”

“…In Section 4, we restrict ourselves to matrices A that are skew-symmetric, in symbols: A = −A T . Tropical eigenvectors of skew-symmetric matrices arise in pairwise comparison ranking, in the approach that was pioneered by Elsner and van den Driessche [6,7]. In [11], the second author offered a comparison with two other methods for statistical ranking, and she noted that the eigenvalue map A → λ(A) for skew-symmetric A is linear on (the cones over) the facets of the cographic zonotope associated with the complete graph on n vertices.…”

On the Markov chain tree theorem in the Max algebra

“…In particular, it applies to symmetrical reciprocal matrices, which arise in the Analytic Hierarchy Process: a widely used framework for multi-criteria decision making [20]. In the spirit of [5,12,13], it would be interesting to investigate the possibility of using the maximal ITW vector in this and related applications. We hope to report on this in more detail in future work.…”