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## Re: [igraph] test network topology

 From: Gabor Csardi Subject: Re: [igraph] test network topology Date: Sat, 10 May 2008 19:26:48 +0200 User-agent: Mutt/1.5.13 (2006-08-11)

```On Fri, May 09, 2008 at 07:59:34PM +0200, Simone Gabbriellini wrote:
[...]
>
> I though that there should be a statistical way to say that my network
> is approximable by a random graph (more than a small world)... but I
> suppose Gabor is right when he says that the only thing I can say is
> that the density of my empiric network is approximable by a random
> distribution...
>
> have I undestood it correctly, Gabor?

Random distribution? I'm a bit puzzled to be honest. What i've tried to
say is that
1) usually you want to compare your system to a "random" system. This
is to see how "random" your network is, or in fact rather the opposite
how much ordered it is compared to a random graph.
2) normally this is done via a statistical test, and then you have a p-value,
the probability that your system "is just random"
3) for graphs we don't really have established statistical tests, not even for
individual graph properties.
4) what we usually do is that we choose a random graph model (Gnm or
configuration
usually) and then say that the examined structural property is not present
in
the random network, thus it is not the consequence of the density (Gnm) or
degree dist. (configuration) of the network.

I give you an example. The forest fire model (forest.fire.game) is known to
generate highly transitive networks. But are these transitive only because
they have a high density?

This question can be answered by generating random ER (Gnm) graphs with the
same number of vertices and edges and measuring their transitivity.

But is the transitivity only a consequence of the degree distribution of the
network?

This question can be answered by generating random graph with the same
degree sequence and then measuring their transitivity.

Both ways measure whether our network is just random or not. But we can only
do this by focusing at a given structural property at a time (transitivity
in the example). Even in this case we don't have the statistical background
to calculate a p-value, which is a pity.

But there might be good statistical tests for networks, i just don't know
about them, e.g. it might be worth to look at 'cugtest' in the 'sna' package.

G.

[...]

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