"Second, we require that the distance be invariant to the number of documents, given any fixed number of clusters in each clustering. Third, we set the scale of the measure by fixing the minimum distance to zero and the maximum distance to logðkÞ. A key point is that none of these axioms requires that one artificially “align” clusterings before judging their distance, as some others have attempted; in fact, we do not even restrict the clusterings to have the same number of clusters."
I don't quite understand why the distance need to be invariant to the number of documents, and how "aligning clusterings" will affect how the distance is judged?
"Second, we require that the distance be invariant to the number of documents, given any fixed number of clusters in each clustering. Third, we set the scale of the measure by fixing the minimum distance to zero and the maximum distance to logðkÞ. A key point is that none of these axioms requires that one artificially “align” clusterings before judging their distance, as some others have attempted; in fact, we do not even restrict the clusterings to have the same number of clusters."
I don't quite understand why the distance need to be invariant to the number of documents, and how "aligning clusterings" will affect how the distance is judged?