Many-to-many Correspondences between Partitions: Introducing a Cut-based Approach

Let $\mathcal{P}$ and $\mathcal{P}'$ be finite partitions of the set $V$. Finding good correspondences between the parts of $\mathcal{P}$ and those of $\mathcal{P}'$ is helpful in classification, pattern recognition, and network analysis. Unlike common similarity measures for partitions that yield only a single value, we provide specifics on how $\mathcal{P}$ and $\mathcal{P'}$ correspond to each other. To this end, we first define natural collections of best correspondences under three constraints \cone, \ctwo, and \cthree. In case of \cone, the best correspondences form a minimum cut basis of a certain bipartite graph, whereas the other two lead to minimum cut bases of $\mathcal{P}$ \wrt $\mathcal{P}'$. We also introduce a constraint, \cfour, which tightens \cthree; both are useful for finding consensus partitions. We then develop branch-and-bound algorithms for finding minimum $P_s$-$P_t$ cuts of $\mathcal{P}$ and thus $\vert \mathcal{P} \vert -1$ best correspondences under \ctwo, \cthree, and \cfour, respectively. In a case study, we use the correspondences to gain insight into a community detection algorithm. The results suggest, among others, that only very minor losses in the quality of the correspondences occur if the branch-and-bound algorithm is restricted to its greedy core. Thus, even for graphs with more than half a million nodes and hundreds of communities, we can find hundreds of best or almost best correspondences in less than a minute.