  Community detection in networks refers to the process of seeking strongly
internally connected groups of nodes which are weakly externally connected. In
this work, we introduce and study a community definition based on internal edge
density. Beginning with the simple concept that edge density equals number of
edges divided by maximal number of edges, we apply this definition to a variety
of node and community arrangements to show that our definition yields sensible
results. Our community definition is equivalent to that of the Absolute Potts
Model community detection method (Phys. Rev. E 81, 046114 (2010)), and the
performance of that method validates the usefulness of our definition across a
wide variety of network types. We discuss how this definition can be extended
to weighted, and multigraphs, and how the definition is capable of handling
overlapping communities and local algorithms. We further validate our
definition against the recently proposed Affiliation Graph Model
(arXiv:1205.6228 [cs.SI]) and show that we can precisely solve these
benchmarks. More than proposing an end-all community definition, we explain how
studying the detailed properties of community definitions is important in order
to validate that definitions do not have negative analytic properties. We urge
that community definitions be separated from community detection algorithms and
propose that community definitions be further evaluated by criteria such as
these.
