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

TL;DR

This paper introduces a cut-based approach to find and analyze many-to-many correspondences between partitions, useful for classification and network analysis, with efficient algorithms capable of handling large graphs.

Contribution

It defines new correspondence constraints, formulates them as minimum cut problems, and develops branch-and-bound algorithms for efficient computation, enabling large-scale applications.

Findings

Efficient algorithms find hundreds of correspondences in large graphs within a minute.

Minor loss in correspondence quality when using greedy algorithms.

Insights into community detection through partition correspondences.

Abstract

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.