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SOCNET  February 2013

SOCNET February 2013

Subject:

Re: Very low pathlength in small-world networks

From:

Julian Hagenauer <[log in to unmask]>

Reply-To:

Julian Hagenauer <[log in to unmask]>

Date:

Thu, 21 Feb 2013 12:37:25 +0100

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*****  To join INSNA, visit http://www.insna.org  *****

Am Thu, 21 Feb 2013 08:20:18 +0000
schrieb Vincenzo Nicosia <[log in to unmask]>:

> On Thu, Feb 21, 2013 at 08:40:10AM +0100, Julian Hagenauer wrote:
> 
> [cut]
> 
> > 
> > So, my questions:
> > a) Is a network with high clustering coefficent and low path length
> > (lower than the one of a random network) still a small-world
> > network? b) Is there some more precise definition of small-world
> > networks?
> > 
> 
> To my knowledge, the definition of small-world has to do with the
> scaling of the typical path length L (also called average shortest
> path length) with the order of the graph N (the number of nodes). For
> a network to be considered a small-world it is necessary that L scales
> as log(N), as happens for Erd"os-Renyi random graphs. So an
> Erd"os-Renyi graph is a small-world, according to this definition,
> even if the clustering coefficient of such graphs vanishes with N
> (precisely, C \propto N^{-1}).
> 
> Apart from that, it is true that in many real networks, especially
> social networks, the small-world property is often associated to a
> relatively high clustering coefficient, and the Watts-Strogatz model
> gives account of this observation in a simple and elegant way. But I
> wouldn't require a network to exhibit high clustering in order to be
> considered a small-world.
> 
> My2Cents
> 
> Vincenzo Nicosia
> 

Thank your for your answer. I basically agree to your point.
However, the log(N)-property cannot be tested for real-world networks,
can it?
So, the question is how to test if a real-world network is a
small-world network. Does comparing the average path lengths to a
random-erdös network with the same number of nodes and edges suffice
for this purpose?

Sincerely
Julian

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