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