Combining Multiple Ranking Systems on the Generalized Permutation Rank Space

Given t ranking systems of n objects, consensus ranking (CR) aims to derive a ranking which best represents the consensus goal of these ranking systems. Since a ranking system of n objects is equivalent to a permutation of the n natural numbers [1, n] from 1 to n, the CR problem has been studied on the bubble-sort graph rank space B
n
which consists of the set of all permutations of order n. However, it remains a challenging issue when a combination of ranking systems (consensus ranking) has ties. But B
n
does not include permutation (or rank systems) with ties. In this paper, we propose a multi-layer combinatorial fusion algorithm for combining multiple ranking systems on the generalized permutation rank space where ties are allowed. Using two simulated data sets and an empirical data set in the molecular docking domain, we demonstrate the robustness of our approach. This study also provides a viable approach to data analytics, machine learning, and combinatorial fusion in the non-parametric rank space.