An edge density definition of overlapping and weighted graph communities

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 and propose that community definitions be further evaluated by criteria such as these.