Understanding the correlation between two different scores for the same set
of items is a common problem in information retrieval, and the most commonly
used statistics that quantifies this correlation is Kendall's τ. However,
the standard definition fails to capture that discordances between items with
high rank are more important than those between items with low rank. Recently,
a new measure of correlation based on average precision has been proposed to
solve this problem, but like many alternative proposals in the literature it
assumes that there are no ties in the scores. This is a major deficiency in a
number of contexts, and in particular while comparing centrality scores on
large graphs, as the obvious baseline, indegree, has a very large number of
ties in web and social graphs. We propose to extend Kendall's definition in a
natural way to take into account weights in the presence of ties. We prove a
number of interesting mathematical properties of our generalization and
describe an O(nlogn) algorithm for its computation. We also validate the
usefulness of our weighted measure of correlation using experimental data