Please find two draft working definitions of "connectivity" and "community network" here https://public.etherpad-mozilla.org/p/DC3_definitionshttps://public.etherpad-mozilla.org/p/DC3_definitions

The connectivity definition is based on THE works of BEREC (the Body of European Regulators for Electronic Communications) as well as on RFC 4084 https://tools.ietf.org/pdf/rfc4084.pdfhttps://tools.ietf.org/pdf/rfc4084.pdf 

The community networks definition is based on the definitions proposed by Parminder, in his email, and by Leandro, in the Guifi paper he kindly shared http://people.ac.upc.edu/leandro/pubs/crowds-guifi-en.pdfhttp://people.ac.upc.edu/leandro/pubs/crowds-guifi-en.pdf