Measuring Similarity between Sets of Overlapping Clusters

The typical task of unsupervised learning is to organize data, for example into clusters, typically disjoint clusters (eg. the \math{K}-means algorithm). One would expect (for example) a clustering of books into topics to present overlapping clusters. The situation is even more so in social networks, a source of ever increasing data. Finding the groups or communities in social networks based on interactions between individuals (a measure of similarity) is an unsupervised learning task; and, groups overlap -- an individual can be a chess player and a violin player, in which case he would interact with members of both these groups. The problem we address is not that of finding the overlapping clusters, but of \emph{comparing} two sets of overlapping clusters. Such a task is the basis for comparing two different clusterings, which is important for comparing algorithms with each other or with a ground truth. From the social network point of view, we are particularly interested in quantifying social group evolution -- how much the social group structure of a social network changed -- by comparing the set of groups at consecutive time intervals. There is significant prior work on comparing sets of disjoint clusters (partitions). When overlap is allowed, the problem becomes considerably more complex owing to the possibility of degeneracies, which we illustrate through examples. We describe three novel definitions of the distance between collections of potentially overlapping sets, and present algorithms for computing those distances. We test our algorithms on diverse data sets: collections composed from social groups in Twitter, Blogosphere and Enron Email data.