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

Well-Connected Communities in Real-World and Synthetic Networks