A paradox in community detection

Recent research has shown that virtually all algorithms aimed at the identification of communities in networks are affected by the same main limitation: the impossibility to detect communities, even when these are well defined, if the average value of the difference between internal and external node degrees does not exceed a strictly positive value, in the literature known as detectability threshold. Here, we counterintuitively show that the value of this threshold is inversely proportional to the intrinsic quality of communities: the detection of well-defined modules is thus more difficult than the identification of ill-defined communities.