Clustering list of list of integers
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I would use the Mutual Information between two sets as the criterion of similarity. In your special case, set the mutual information of those sets whose intersection is less than X% to $0$. Run your clustering and see what comes out. With the same similarity matrix, you can also construct a weighted graph and cluster nodes (which is called Community Detection). This link gives you a proper overview starting from slide 37 and in slide 48 you find your own idea! If you have more questions about Community Detection, e.g. implementation, please drop a comment here.