Finding related-node clusters
The problem you are trying to solve is called community detection or community structure, where several popular algorithms are listed, such as Girvan–Newman algorithm and Minimum-cut method.

You can find many related questions and answers on this site, on CrossValidate and on StackOverflow, such as the following:


Community detection in weighted directed graphs for fixed number of communities.
How to do community detection in a weighted social network/graph?
What are the differences between community detection algorithms in igraph?


A good introduction can be found at Community detection in graphs.

A recent survey can be found at Community detection in social networks.