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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

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