Lecture 9: Rank Aggregation in MetaSearch MetaSearch Engine Social Choice Rules Rank Aggregation Choices of Search Engines Many search engines exist to compete for users The results are not necessarily the same Different users prefer different search engines
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How do we make decisions
in a group with
1 2 3
1stX YX 1 : X>Y
Pref. i.e.Person 2 : Y>X
2ndYXY 3 : X>Y
How to Aggregate their preferences? How to choose?
1 2 3
1stX YZ 1 : X>Y>Z
Pref. 2ndYZ Xi.e. Person 2 : Y>Z>X
3rdZXY 3 : Z>X>Y
1 2 3
1stX YZ 1 : X>Y
Pref. 2ndYZ X for (x,y) 2 : Y>X X>Y
3rdZXY 3 : X>Y
Person alternative can win a majority against all other alternatives.
1 2 3
1stX(1) Y(1)X(1) X(4) X
Pref. 2ndY(2)X(2) W(2) Y(7) Y
3rdZ(3)W(3)Z(3) Z(10) W
4thW(4)Z(4)Y(4) W(9) Z
Then We get choice X>Y>W>Z
X>Y in 2 out of 3, Y>W in 2 out of 3,
W>Z in 2 out of 3, X>W in 3 out of 3,
X>Z in 3 out of 3, Y>Z in 2 out of 3
Then we can achieve same choice
1 with relevant scores available
allotting each input system a point p to be distributed according to relevance scores of the documents.
2 Weighted Borda-rule
Each voter may not have equal effectiveness to the final result. We may set more weight to good quality input systems.
It also comes from social choice theory. The Condorcet algorithm says that any candidate that can beat all other candidates in a head-to-head contest (pair-wise comparison) should win the election.
For each candidate pair (x,y), there exists an edge from x to y if x would receive at least as many votes as y in a head-to-head contest.
In Condorcet graph, there is at least one directed edge between every pair of candidates. ( we call the graph is semi-complete)
It may contains cycles in the graph. This is due to voting paradox of the condorcet voting.
Definition of Strongly connected component(SCC):
A strongly connectedsubgraph, S, of a directed graph, D, such that no vertex or subset of vertices of D can be added to S such that the new subgraph is still strongly connected.
The graph is totally orderable at the level of the SCC via binary/paired comparison, we get a tie between all three alternatives with the Borda’s and each SCC is a “pocket” of cycles, within which each candidate is tied.
Step 3, The condorcet-consistent Hamiltonian path is any Hamiltonian path through Condorcet graph.
Definition Hamiltonian path: A path between two vertices of a graph that visits each vertex exactly once.
Refer to[Javed A. Aslam, Mark Montague 2001] for proof.
Here we discussed two cases which using algorithm rooted at social choice theory for MetaSearch rank aggregation.
[Javed A. Aslam, Mark Montague 2001] Models for Metasearch
[Cynthia Dwork, Ravi Kumar, Moni Naor, D.Sivakumar 2001]
Rank Aggregation Methods for the Web in WWW10
[Fox and Shaw,1995]
The final score of each document d is based on the scores given to d by each input systems (voters).
Algorithm Final score
CombMin minimum of individual relevance scores
CombMed median of individual relevance scores
CombMax maximum of individual relevance scores
CombSum sum of individual relevance scores
CombANZ CombSum / num non-zero relevance scores
CombMNZ CombSum * num non-zero relevance scores
The final score of document d is a simply linearly (each weighted differently) combining the normalized relevance scores given to each document.
New Challenges: Different from the case in TREC data fusion,
Given several full lists , for instance, the normalized Footrule distance of to is given by
If are partial lists, let U denote the union of elements in and let be a full list with respect to U. Considering the distance between and the projection of with respect to , we have the induced footrule distance
The question is
Given (full or partial) lists , find a such that is a full list with respect to the union of the elements of
The aggregation obtained by optimizing Kendall distance is called Kemeny optimal aggregation.
(please refer to [Cynthia Dwork, Ravi Kumar, Moni Naor, D.Sivakumar 2001] for detailed proof )
We can use Spearman footrule distance to approximate the Kendall distance.
It has following characteristics. is a procedure that computes a locally Kemeny optimal aggregation of that is maximally consistent with .