  We have recently introduced a multistep extension of the greedy algorithm for
modularity optimization. The extension is based on the idea that merging l
pairs of communities (l>1) at each iteration prevents premature condensation
into few large communities. Here, an empirical formula is presented for the
choice of the step width l that generates partitions with (close to) optimal
modularity for 17 real-world and 1100 computer-generated networks. Furthermore,
an in-depth analysis of the communities of two real-world networks (the
metabolic network of the bacterium E. coli and the graph of coappearing words
in the titles of papers coauthored by Martin Karplus) provides evidence that
the partition obtained by the multistep greedy algorithm is superior to the one
generated by the original greedy algorithm not only with respect to modularity
but also according to objective criteria. In other words, the multistep
extension of the greedy algorithm reduces the danger of getting trapped in
local optima of modularity and generates more reasonable partitions.
