Comparing network covers using mutual information

In network science, researchers often use mutual information to understand the difference between network partitions produced by community detection methods. Here we extend the use of mutual information to covers, that is, the cases where a node can belong to more than one module. In our proposed solution, the underlying stochastic process used to compare partitions is extended to deal with covers, and the random variables of the new process are simply fed into the usual definition of mutual information. With partitions, our extended process behaves exactly as the conventional approach for partitions, and thus, the mutual information values obtained are the same. We also describe how to perform sampling and do error estimation for our extended process, as both are necessary steps for a practical application of this measure. The stochastic process that we define here is not only applicable to networks, but can also be used to compare more general set-to-set binary relations.