Refinements

The Basic-CLICK algorithm will divide the graph into kernels and singletons. To use the algorithm for solving clustering problems, we must introduce a number of refinements:

• Removing Negative Edges: The MINCUT problem for a weighted graph with both positive and negative edges is NP-Complete. In order to use the efficient MINCUT algorithms we must remove the negative edges. By modifying the algorithm slightly, we can still use the new graph to find kernels in the original graph.
• Adoption Step: In practice, ``true'' clusters are usually larger than just the kernel. To accomodate this, in the refined algorithm, kernels ``adopt'' singletons to create larger clusters. This is done by searching for a singleton v and a kernel K, whose pairwise fingerprint similarity is maximum among all pairs of singletons and kernels. The refined algorithm iteratively applies the adoption step and then the Basic-CLICK algorithm on the remaining singletons, stopping when there are no more changes.
• Merge Step: In this step we merge clusters whose fingerprints are similar (the justification for this is that, in practice, clusters can contain multiple kernels). The merging is done iteratively, each time merging two clusters whose fingerprint similarity is the highest (provided that the similarity exceeds a predefined threshold).