Rank Centrality: Ranking from Pair-wise Comparisons (1209.1688v4)

Abstract: The question of aggregating pair-wise comparisons to obtain a global ranking over a collection of objects has been of interest for a very long time: be it ranking of online gamers (e.g. MSR's TrueSkill system) and chess players, aggregating social opinions, or deciding which product to sell based on transactions. In most settings, in addition to obtaining a ranking, finding `scores' for each object (e.g. player's rating) is of interest for understanding the intensity of the preferences. In this paper, we propose Rank Centrality, an iterative rank aggregation algorithm for discovering scores for objects (or items) from pair-wise comparisons. The algorithm has a natural random walk interpretation over the graph of objects with an edge present between a pair of objects if they are compared; the score, which we call Rank Centrality, of an object turns out to be its stationary probability under this random walk. To study the efficacy of the algorithm, we consider the popular Bradley-Terry-Luce (BTL) model (equivalent to the Multinomial Logit (MNL) for pair-wise comparisons) in which each object has an associated score which determines the probabilistic outcomes of pair-wise comparisons between objects. In terms of the pair-wise marginal probabilities, which is the main subject of this paper, the MNL model and the BTL model are identical. We bound the finite sample error rates between the scores assumed by the BTL model and those estimated by our algorithm. In particular, the number of samples required to learn the score well with high probability depends on the structure of the comparison graph. When the Laplacian of the comparison graph has a strictly positive spectral gap, e.g. each item is compared to a subset of randomly chosen items, this leads to dependence on the number of samples that is nearly order-optimal.

• The paper introduces the Rank Centrality algorithm, which translates pair‐wise comparisons into a global ranking through a random walk framework.
• The paper establishes theoretical error bounds under the Bradley-Terry-Luce model, demonstrating near-optimal performance in graphs with a positive spectral gap.
• The paper validates Rank Centrality using synthetic experiments, showing comparable accuracy to maximum likelihood estimators and improved efficiency over existing methods.

Overview of "Rank Centrality: Ranking from Pair-wise Comparisons"

The paper Rank Centrality: Ranking from Pair-wise Comparisons introduces a novel iterative algorithm named Rank Centrality that aggregates pair-wise comparisons to derive a global ranking for a set of objects. This problem is prevalent across various domains, including online gaming, chess, social preference aggregation, and product recommendation systems. Beyond ranking, there is an interest in determining the scores for each object, which provides insight into the intensity of preferences.

Key Contributions

1. Random Walk Interpretation: The Rank Centrality algorithm is derived from a random walk on a graph where nodes represent objects, and edges represent pair-wise comparisons. The score, termed Rank Centrality, corresponds to the stationary distribution of this walk.
2. Theoretical Framework: The algorithm's effectiveness was evaluated under the Bradley-Terry-Luce (BTL) model, which is synonymous with the Multinomial Logit (MNL) model for pair-wise comparisons. The paper provides bounds on the finite sample error rates, demonstrating that the sample complexity to learn scores depends on the graph's structure and spectral properties.
3. Experimental Validation: Experiments on synthetic datasets generated from the BTL model reveal that Rank Centrality performs comparably to the Maximum Likelihood (ML) estimator and exceeds the performance of other leading ranking algorithms.

Noteworthy Findings

• Spectral Gap Influence: The results underscore that the algorithm's performance is nearly optimal when pair-comparisons form a graph with positive spectral gap, i.e., when items are compared to random subsets. The spectral gap is crucial for ensuring that the random walk mixes well and converges to the true score distribution efficiently.
• Error Bounds: The paper details that with a sample size on the order of $O(n\log n)$, the error in the rank estimation diminishes significantly. This showcases the efficacy of Rank Centrality in scenarios where large-scale comparisons are infeasible.

Implications and Future Directions

The introduction of Rank Centrality provides a computationally efficient, theoretically grounded method for ranking objects based on incomplete pair-wise comparisons. The implications are substantial for domains relying on aggregated rankings from noisy, conflicting data. Furthermore, the paper suggests that future investigations could enhance the algorithm's robustness through increased regularization techniques and exploring alternative spectral ranking methods. Potential developments might also delve into more adaptable algorithms that dynamically select informative pairs, thereby refining the estimation process with fewer comparisons.

Conclusion

Rank Centrality: Ranking from Pair-wise Comparisons contributes significantly to the literature on ranking algorithms by offering a practical, robust, and theoretically sound solution that addresses the complex nature of aggregating pair-wise comparisons. The use of random walks and spectral properties marks a distinct advance, aligning well with both theoretical expectations and practical requirements across varied applications. The findings open avenues for further research into adaptive algorithms and broader applicability in domains requiring efficient ranking from limited data.