The issue of partitioning a network into communities has attracted a great
deal of attention recently. Most authors seem to equate this issue with the one
of finding the maximum value of the modularity, as defined by Newman. Since the
problem formulated this way is NP-hard, most effort has gone into the
construction of search algorithms, and less to the question of other measures
of community structures, similarities between various partitionings and the
validation with respect to external information. Here we concentrate on a class
of computer generated networks and on three well-studied real networks which
constitute a bench-mark for network studies; the karate club, the US college
football teams and a gene network of yeast. We utilize some standard ways of
clustering data (originally not designed for finding community structures in
networks) and show that these classical methods sometimes outperform the newer
ones. We discuss various measures of the strength of the modular structure, and
show by examples features and drawbacks. Further, we compare different
partitions by applying some graph-theoretic concepts of distance, which
indicate that one of the quality measures of the degree of modularity
corresponds quite well with the distance from the true partition. Finally, we
introduce a way to validate the partitionings with respect to external data
when the nodes are classified but the network structure is unknown. This is
here possible since we know everything of the computer generated networks, as
well as the historical answer to how the karate club and the football teams are
partitioned in reality. The partitioning of the gene network is validated by
use of the Gene Ontology database, where we show that a community in general
corresponds to a biological process.Comment: To appear in Physica A; 25 page