  A measure of distance between two clusterings has important applications,
including clustering validation and ensemble clustering. Generally, such
distance measure provides navigation through the space of possible clusterings.
Mostly used in cluster validation, a normalized clustering distance, a.k.a.
agreement measure, compares a given clustering result against the ground-truth
clustering. Clustering agreement measures are often classified into two
families of pair-counting and information theoretic measures, with the
widely-used representatives of Adjusted Rand Index (ARI) and Normalized Mutual
Information (NMI), respectively. This paper sheds light on the relation between
these two families through a generalization. It further presents an alternative
algebraic formulation for these agreement measures which incorporates an
intuitive clustering distance, which is defined based on the analogous between
cluster overlaps and co-memberships of nodes in clusters. Unlike the original
measures, it is easily extendable for different cases, including overlapping
clusters and clusters of inter-related data for complex networks. These two
extensions are, in particular, important in the context of finding clusters in
social and information networks, a.k.a communities.
