About Newman and Girvan definition of Modularity of Network-Community

Question

Can some body explain me the following statement about Community Structure in a Graph : Newman and Girvan write: „In a network in which edges fall between vertices without regard for the communities they belong to, we would have :

I mean : what is meant by 'without regard for the communities they belong to', since here is a community ! itself, so why he said without regard for the communities they belong to. I suggest he should said: 'with regard for the communities they belong to'--> not [without] --<

Am I right ?

Answer 1

I think that the authors means "without". They wanted to test the modularity measure in networks where there is no community structure, i.e, for random structure of the networks.

Answer 2

This quote and equation could not be located in Girvan and Newman's work. It seems to come from an article called "Modularity for Bipartite Networks" authored by Tsuyoshi Murata published in a book called "Data Mining for Social Network Data" by Memon et al.

The $a_i$ are defined as row sums over the $e_{ij}$ in an unnumbered equation before the sentence, so they are akin to the marginal probabilities of $i$, and the statement that $e_{ij}=a_i a_j$ seems to be akin to the statement that the probability of having an edge between vertices $i$ and $j$ is independent of the community, so the correct wording is surely "without" here.