# WIRE: A Weighted Item Removal Method for Unsupervised Rank Aggregation
## 4 WIRE: Algorithm Design and Analysis  
**Algorithm description. ** Most weighted rank aggregation methods integrate the learned weights into the scores of the individual elements, with the sole objective of improving the quality of their produced ranking. As mentioned earlier, a representative exception is the list pruning policy of [13], which uses the learned ranker weights to remove the weakest elements from the input lists. In the sequel, we introduce a new method called _WIRE (Weighted Item Removal)_ that utilizes the learned list weights to effectively determine the contribution of each input preference list in the formulation of the output list \(\mathcal{L}\). WIRE introduces a novel mechanism that identifies appropriate list elements for removal. Notably, this mechanism depends only on the input preference lists, the ranker weights, and the consensus ranking. This increases the flexibility of our algorithm, since its independence from other factors (e.g., the scores of the input list elements) enables its attachment to any weighted rank aggregation method \(\mathcal{A}_{w}\) as a post-processing step. On the other hand, if \(\mathcal{A}_{w}\) cannot estimate the ranker weights in a robust manner, then the ability of WIRE to identify the weakest elements will inevitably decrease. For this reason, we present WIRE in the context of DIBRA, the distance-based weighted method that was introduced in [13]. The experiments of [13] have shown that DIBRA is superior to other weighted algorithms (e.g., [26; 27]), and this was also verified in the present experiments. Figure 1: Schematic diagram of weighted rank aggregation with list pruning. Therefore, even though our method can be applied in combination with any weighted rank aggregation method, the capability of DIBRA in distinguishing the expert rankers from the non-experts renders it a suitable basis for applying WIRE. In short, DIBRA capitalizes on the concept that the importance of a ranker is determined by the distance of its preference list from the consensus ranking \(\mathcal{L}\). In this spirit, it iteratively adjusts the weight \(w^{(u)}\) of a ranker \(u\) by using the following equation:  
\[w^{(u)}_{i}=w^{(u)}_{i-1}+\exp\Big{(}-i\cdot d\big{(}l^{(u)},\mathcal{L}_{i-1} \big{)}\Big{)}, \tag{5}\]  
where \(w^{(u)}_{i}\) and \(\mathcal{L}_{i}\) are the weight of ranker \(u\) and the aggregate list after \(i\) iterations, respectively. Moreover, \(d\) represents a function that quantifies the distance between the preference list \(l^{(u)}\) and the aggregate list \(\mathcal{L}\). The form of Equation (5) guarantees that the weights converge after several iterations, usually between 5 and 20. After the voter weights have been calculated and the final aggregate list \(\mathcal{L}\) has been constructed, WIRE is deployed to further improve the quality of \(\mathcal{L}\). The proposed method refines the input preference lists by removing their weakest elements and operates in five phases, according to Algorithm 1. ```
1:Rankers \(u_{1}\ldots u_{n}\), Ranker weights \(w^{(u_{1})},\ldots,w^{(u_{n})}\), Preference lists \(l^{(u_{1})},\ldots,l^{(u_{n})}\), number of buckets \(B<n\), hyper-parameter \(\delta_{1}\)
2:Output: Aggregate list \(\mathcal{L}^{\prime}\).