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ad 1. The matrix eii will be symmetric because the network
Newman&Girvan analyze is undirected. The assortativity coefficient on
which Q is based (see ref. 34 in N&G paper you refer to) was in fact
developed for directed networks, so that eii does not have to be
ad 2. ai*ai make sense as a null model because it corresponds to
"proportionate mixing": fraction of ties between groups i and j is
proportional to the number of links they have. See Koehly, L. M.,
Goodreau, S. M., Morris, M., 2004. Exponential family models for
sampled and census network data. Sociological Methodology 34, 241–270
for a nice discussion of possible null models for mixing matrices.
Your null model corresponds to one of the conditional log-linear
models discussed in section 3.1 of that paper.
The term "proportionate mixing", as far as I know, comes from Nold,
A., 1980. Heterogeneity in disease-transmission modelling.
Mathematical Biosciences 52 (3–4), 227–240.
I hope this helps.
On Tue, Jun 4, 2013 at 3:27 PM, Kropp Per <[log in to unmask]> wrote:
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> Hello everybody,
> I use a formula from Newman/Girvan (Phys. Rev. E 69 (2004) 026113) to
> calculate Q=SUMi(eii-ai*ai) with ai=SUMj(eij) from the k x k symmetric
> matrix e. eii is the fraction of links within the cluster k and ai is the
> row (or column) sum. Very easy to implement in standard statistical
> packages. Here are my questions:
> 1. Why has matrix e to be symmetric?
> 2. Why can ai*ai be characterized as an appropriate null model?
> (Both are questions from a reviewer ;-))
> Best regards to the specialists
> Dr. Per Kropp
> Institute for Employment Research
> Regional Research Network
> Frau-von-Selmnitz-Str. 6, D-06110 Halle
> Phone: +49/345-1332-236 (secretary: -255; fax: -555)
> mail: [log in to unmask]
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