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I'm trying to find a fixed number of communities in a fully connected graph with weighted edges.

How to do community detection in a weighted social network/graph? suggests the use of iGraph's fastgreeedy function for community detection.

I have a graph with vertices as people and edge weights as the similarity between the vertices.

I'm new to graph theory so I'm not sure if this is correct but I'm in a way trying to achieve max-sum k-clustering.

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This post discusses two possible igraph community functions that allow you to set a specific number of communities:

https://stackoverflow.com/a/38899957/1333650

Quoting from Jim Leach:

You can use either the cluster_spinglass() function and set spins to be the number of communities desired. Or use one of the hierarchical methods and then use cut_at() to get the desired number of communities, using the no argument to specify how many communities you want. [....] Note that the spinglass method will give you back a communities object, whereas the cut_at method simply gives you back the community indices for all nodes in the graph (i.e. a simple numeric vector).

However, unless there is a specific network metric that you want to inform the formation of your communities, you might just want to use a normal k-means clustering algorithm directly on the weight adjacency matrix, something like this:

https://stat.ethz.ch/R-manual/R-devel/library/stats/html/kmeans.html

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