Twitter Data-Analyse: What can I do with the data?
Community analysis implies graph analysis.
here is a short list of things you can work on:

People often reshares tweets among a certain social group. Minimum-cut method, Girvan–Newman and Modularity maximization are someof the starting algorithms to extract these type of substructures.
You can try and find different hierarchies among the groups sharing a particular topics
You can try and analyse the lifetime of tweets for particular topics (survival analysis)

Analysing tweets is closer to graph analytics rather than NLP.
Here is a great overview on community analysis. For coding and algorithms, please check graphX Spark library. If your data is not too large, networkX is easier.
For survival analysis, lifeline is one of the easier options.