What is the difference between the useage of the word "operate" and "run"?

While both words can be used, "to operate" would usually be used when speaking of a physical interaction. For example, someone operates a machine. "To run", however, would be used when synonyms like: "to manage", "to control" or "to be in charge of" can be applied. Considering your two options, "to run" would be more natural for the managing of a community.