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enter image description hereI currently have an undirected weighted graph. I am trying to find a good algorithm for learning/finding a cluster of closely related nodes and/or loosely relationship within a cluster in a graph given a size of the cluster. For example, here the longer edges represent the farther relationship between 2 nodes. We identified 2 clusters where each cluster has a close relationship internally. I am new to computer science, could you please suggest an algorithm or a paper for this problem? Your help is really appreciated!

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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:

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

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

John L.
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