Condensation of the digraph associated with a reciprocal matrix and a vector

Abstract

Reciprocal matrices are a fundamental tool in the Analytic Hierarchy Process (AHP), where priority vectors are typically derived from pairwise comparisons. The efficiency of a positive vector, in the sense of Pareto optimality, can be characterized through the strong connectivity of a directed graph $G_{A,w}$ associated with a reciprocal matrix $A$ and a vector $w$. In this paper, we investigate the structure of the condensation digraph of $G_{A,w}$ in the case where $w$ is inefficient, with particular emphasis on the Perron vector. We provide a characterization of this structure and derive several constructive results. In particular, we show how efficiency can be achieved by modifying a single pair of reciprocal entries, and we construct an augmented reciprocal matrix whose efficient vector extends $w$. Finally, we propose a procedure to transform $w$ into an efficient vector for $A$.