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Networks represent how the entities of a system are connected and can bepartitioned differently, prompting ways to compare partitions. Commonapproaches for comparing network partitions include information-theoreticmeasures based on mutual information and set-theoretic measures such as theJaccard index. These measures are often based on computing the agreement interms of overlap between different partitions of the same set. However, theyignore link patterns which are essential for the organisation of networks. Wepropose flow divergence, an information-theoretic divergence measure forcomparing network partitions, inspired by the ideas behind the Kullback-Leiblerdivergence and the map equation for community detection. Similar to theKullback-Leibler divergence, flow divergence adopts a coding perspective andcompares two network partitions $\mathsf{M}_a$ and $\mathsf{M}_b$ byconsidering the expected extra number of bits required to describe a randomwalk on a network using $\mathsf{M}_b$ relative to reference partition$\mathsf{M}_a$. Because flow divergence is based on random walks, it can beused to compare partitions with arbitrary and different depths. We show thatflow divergence distinguishes between partitions that traditional measuresconsider to be equally good when compared to a reference partition. Applied toreal networks, we use flow divergence to estimate the cost of overfitting inincomplete networks and to visualise the solution landscape of networkpartitions.

Flow Divergence: Comparing Maps of Flows with Relative Entropy