Application of hodge theory to the analysis of ranking data - Laurent, Valerie Evangelin (2019)
This paper presents a thorough analysis of a ranking method called HodgeRank. It is derived from Hodge theory, in particular Hodge decomposition. The objective of this paper is to study the theoretical framework of HodgeRank, to build the ranking model of HodgeRank and to apply HodgeRank to the real-world ranking problems. HodgeRank uses pairwise ranking approach and its ranking model is built on a network graph. HodgeRank produces the global ranking of the given set of alternatives and the measurement of the reliability of the global ranking. The algorithm of HodgeRank consists of three major parts, which are the formulation of the pairwise ranking, the solution to the optimization problem to find the global ranking, and the decomposition of the pairwise ranking to measure the reliability of the global ranking. HodgeRank stands out from other ranking method due to its flexibility and adaptability. Not only it can work with an incomplete and imbalance ranking data, but also it can be easily modified to suit the environment of the ranking problem. Four HodgeRank applications are provided to illustrate the basic idea of the method and the construction of the ranking model. The first application aims to find the global ranking of a set of movies. The second application aims to detect forex arbitrage in currency market. The third application aims to rank protein structures from its unfolded state to its folded state. The fourth application aims to rank topologically association domains based on its complexity.