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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 because of the assumption about the degree distribution
(Poisson). Newer 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
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.
& 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
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