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Dear colleagues,

One colleague -- let me honour her by mentioning her name: Stasa Milojevic,
Indiana University -- noted that the average degree in a random network
[d(rg)] scales with the logarithm of the number of vertices, but d(rg) =
log(n)/ log(z) and not the other way round. This solves the problem!

It was also noted that the Erdos-Renyi model provides a poor approximation
models are available (Newman et al., 2002, PNAS). However, for my case, E-R
does the job: it provides results which tend in the same direction as the
analytical solution: I have a small-world effect in my data after 2004.

Best wishes,
Loet

_____

Loet Leydesdorff
Amsterdam School of Communications Research (ASCoR),
Kloveniersburgwal 48, 1012 CX Amsterdam.
Tel.: +31-20- 525 6598; fax: +31-842239111
<http://www.leydesdorff.net/> http://www.leydesdorff.net/

_____

Sent: Wednesday, July 28, 2010 8:17 AM
Subject: testing for the small-world effect

Dear colleagues,

In my current research project, I have theoretical reasons to believe that
an oligopoly is shaped over time, and therefore I would like to test the
(network) data for different years on the small-world effect. My colleague
and coauthor suggested two ways to do so, but they lead to very different
conclusions.

1. Using Pajek, for example, one can test the network under Net > Random
Networks > Erdos-Renyi, and use the average degree (found under Info >
Network) as input. The resulting random network can be analyzed in terms of
the clustering coefficient and the average distance. These two parameters
can then be compared with the ones of the empirical network.

Following Watts & Strogatz (1998), Newman et al. (2006, at pp. 288f.)
specified that a small world presumes that (i) the mean vertex-vertex
distance in the empirical network is comparable with that of a random graph
(d / d(rg) ~ 1) and (ii) the clustering coefficient is empirically much
larger. Walsh's (1999) proximity ratio [(CC/CC(rg)) / (d / d(rg)] would then
be much larger than one for a network that contains the small world
property. (I find in my case a ratio > 100; hence, a small world effect).

2. Newman et al. (2006, at pp. 286f.) also suggests an analytical approach
to the problem. CC(rg) would then be z/n -- z being the average degree and n
the number of nodes. Unlike the CC, the mean of the vertex-vertex distances
scales with the logarithm of the number of nodes, and therefore it seems to
follow: d(rg) = log(z) / log(n). The theoretical values can again be
compared with the empirical one. In this case, the Walsh's ratio is appr.
0.8. Hence, there would be no small-world effect in my data.

Did anyone struggle with this problem before and have some advice?
Thank you so much in advance.

Best wishes,
Loet

_____

Loet Leydesdorff
Amsterdam School of Communications Research (ASCoR)
Kloveniersburgwal 48, 1012 CX Amsterdam.
Tel. +31-20-525 6598; fax: +31-842239111