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We prove the incompatibility of certain desirable properties of communitypartition quality functions. Our results generalize the impossibility result of[Kleinberg 2003] by considering sets of weaker properties. In particular, weuse an alternative notion to solve the central issue of the consistencyproperty. (The latter means that modifying the graph in a way consistent with apartition should not have counterintuitive effects). Our results clearly showthat community partition methods should not be expected to perfectly satisfyall ideally desired properties.  We then proceed to show that this incompatibility no longer holds whenslightly relaxed versions of the properties are considered, and we provide infact examples of simple quality functions satisfying these relaxed properties.An experimental study of these quality functions shows a behavior comparable toestablished methods in some situations, but more debatable results in others.This suggests that defining a notion of good partition in communities probablyrequires imposing additional properties.

Incompatibility boundaries for properties of community partitions