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Many community identification methods use Modularity defined by Mark Newman. However, modularity and "modularity picture" cannot incorporate highly Overlapping community structure.

It is considered that networks have hierarchical, or multi-scale community structure. There are some methods (See Link community or other papers^1).

Check out Paul Adams's slides and Adrien Friggeri's slides regarding real-world social community structure.

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Many networks have directional edges. Examples are: emails, WWW, citation, Food web, softwares, ... How do we find communities in the directed networks? What does a community mean in directed networks?

The concept of modularity has a deep connection to the Random walk, and thus it is easily generalizable to directed networks. Elizabeth Leicht and Mark Newman generalized the [modularity]

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