Mathematical Programming

, Volume 127, Issue 1, pp 203–244

Statistical ranking and combinatorial Hodge theory

Open Access
Full Length Paper Series B

DOI: 10.1007/s10107-010-0419-x

Cite this article as:
Jiang, X., Lim, LH., Yao, Y. et al. Math. Program. (2011) 127: 203. doi:10.1007/s10107-010-0419-x


We propose a technique that we call HodgeRank for ranking data that may be incomplete and imbalanced, characteristics common in modern datasets coming from e-commerce and internet applications. We are primarily interested in cardinal data based on scores or ratings though our methods also give specific insights on ordinal data. From raw ranking data, we construct pairwise rankings, represented as edge flows on an appropriate graph. Our statistical ranking method exploits the graph Helmholtzian, which is the graph theoretic analogue of the Helmholtz operator or vector Laplacian, in much the same way the graph Laplacian is an analogue of the Laplace operator or scalar Laplacian. We shall study the graph Helmholtzian using combinatorial Hodge theory, which provides a way to unravel ranking information from edge flows. In particular, we show that every edge flow representing pairwise ranking can be resolved into two orthogonal components, a gradient flow that represents the l2-optimal global ranking and a divergence-free flow (cyclic) that measures the validity of the global ranking obtained—if this is large, then it indicates that the data does not have a good global ranking. This divergence-free flow can be further decomposed orthogonally into a curl flow (locally cyclic) and a harmonic flow (locally acyclic but globally cyclic); these provides information on whether inconsistency in the ranking data arises locally or globally. When applied to statistical ranking problems, Hodge decomposition sheds light on whether a given dataset may be globally ranked in a meaningful way or if the data is inherently inconsistent and thus could not have any reasonable global ranking; in the latter case it provides information on the nature of the inconsistencies. An obvious advantage over the NP-hardness of Kemeny optimization is that HodgeRank may be easily computed via a linear least squares regression. We also discuss connections with well-known ordinal ranking techniques such as Kemeny optimization and Borda count from social choice theory.


Statistical ranking Rank aggregation Combinatorial Hodge theory Discrete exterior calculus Combinatorial Laplacian Hodge Laplacian Graph Helmholtzian HodgeRank Kemeny optimization Borda count 

Mathematics Subject Classification (2000)

68T05 58A14 90C05 90C27 91B12 91B14 
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Copyright information

© The Author(s) 2010

Authors and Affiliations

  • Xiaoye Jiang
    • 1
  • Lek-Heng Lim
    • 2
  • Yuan Yao
    • 3
  • Yinyu Ye
    • 4
  1. 1.Institute for Computational and Mathematical EngineeringStanford UniversityStanfordUSA
  2. 2.Department of MathematicsUniversity of CaliforniaBerkeleyUSA
  3. 3.School of Mathematical Sciences, LMAM and Key Lab of Machine Perception (MOE)Peking UniversityBeijingPeople’s Republic of China
  4. 4.Department of Management Science and EngineeringStanford UniversityStanfordUSA

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