Hi Ettore (or ciao),
Welcome to the forum!
Vincent is the first author of Leiden, so he knows the inner workings of Louvain, smart local move, faster versions thereof, and eventually Leiden. He’s on holiday by will be back in a few days.
Since I have a little knowledge too, I’ll attempt to answer you in the meanwhile. I’m not an R user myself, using Python instead, but am happy to help if I can.
First, why are you looking at Louvain in the first place? It’s an old, slow, and flawed algorithm, which is the whole point of the later improved algos. Do you have an actual graph you are trying to cluster and have observed different results with R-igraph and another library?
Second, the original paper and the Leiden paper more recently, but also the SLM paper from 2013 are pretty good at describing how the clustering happens including the first stage. I myself started there and then went down into the Leiden code without too much trouble. I’m not sure if you need an R-specific explanation of the algorithm, R developers should read those papers just fine.
Third, if you have an actual graph you are working on it would be simpler if you could just share it with us, and also the expected result for the clustering and related modularity. That way we could see if it’s an inconsistency in the implementation.
Cheers,
Fabio