  Integral to the problem of detecting communities through graph clustering is
the expectation that they are "well connected". In this respect, we examine
five different community detection approaches optimizing different criteria:
the Leiden algorithm optimizing the Constant Potts Model, the Leiden algorithm
optimizing modularity, Iterative K-Core Clustering (IKC), Infomap, and Markov
Clustering (MCL). Surprisingly, all these methods produce, to varying extents,
communities that fail even a mild requirement for well connectedness. To
remediate clusters that are not well connected, we have developed the
"Connectivity Modifier" (CM), which, at the cost of coverage, iteratively
removes small edge cuts and re-clusters until all communities produced are well
connected. Results from real-world and synthetic networks illustrate a tradeoff
users make between well connected clusters and coverage, and raise questions
about the "clusterability" of networks and models of community structure.
