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We should check the process getting [ln(n)/ln(avg.degree) ~ average
geodesic].
I referenced

Mark Newman - Random Graphs as Models of Networks
http://www.santafe.edu/research/publications/wpabstract/200202005
.

z = avg. degree

1. the mean number of first neighbors of a vertex = z
the mean number of second neighbors of a vertex ~ z^2
the mean number of mth neighbors of a vertex ~ z^m

2. I quote from upper article.

"When the total number of vertices within distance m is equal to the
size n of the whole graph, m is equal to the so-called "radius" rof the
network around vertex A. ...omited... r (=radius) is thus also approximately
equal to the average vertex-vertex distance L".

3. z^m ~ n => m ~ ln(n)/ln(z)

Every steps used some rough approximations (as like many statistical
physics' applications)
The "mean" in step 1 is rough approximation (the reference seemed to say
nothing about that's accurate distribution. Only mean was mentioned.)
In step 2, 3 there is a clear approximation.

Due to many approximations, maybe the given equation seems to be valid when
'order (rough value)' is considered, not 'accurate value'.

Maybe I could miss something. If there are mistakes, let me know.

Best Regards.
Kim Se Kwon.

----- Original Message -----
Sent: Thursday, December 09, 2004 6:38 PM
Subject: Re: avg. geodesic

> *****  To join INSNA, visit http://www.insna.org  *****
>
> Hi
>
> I made a few experiments myself. I created in Pajek 5 Random Erdos-Renyi
> Undirected General Graphs with n=100, and avg. degree=3, and calclulated
> their avg. geodesic distance with Pajek>Net>Paths between 2
> vertices->Distribution of Distances->from all vertices. Results range from
> 3.69 to 4.62, i.e. they hover around the predicted 4.191 with quite some
> variability.
>
> Then I did the same for the network parameters I am interested in, i.e.
> n=418 avg. degree=87. For all 5 networks I get an avg. geodesic of
> 1.79etc, which is consistently different from the predicted 1.35.
> Could it be that the ln(n)/ln(avg.degree) formula applies only to sparse
> networks?
>
>
> Thanks,
>
> Gianluca
>
>
> > Did you make a Random Erdos-Renyi Undirected General Graph?
> >
> > The approximation you gave may be applied to only random ER graph.
(there
> > are more general approxiations that can be applied more general graph).
> >
> > And you get the average distance of graph using Net->Paths between 2
> > vertices->Distribution of Distances->from all vertices menu.
> >
> > I did two experiments.
> >
> > 1. 100 nodes, average degree 3.
> >     approximation : ln(100)/ln(3)=4.191, real : 4.41981
> > 2. 1000 nodes, average degree 5.
> >     approximation : ln(1000)/ln(5)=4.292, real : 4.48113
> >
> > Is it almost same?
> >
> > Best.
> >
> > ----- Original Message -----
> > Sent: Tuesday, December 07, 2004 9:07 PM
> >
> >
> > > *****  To join INSNA, visit http://www.insna.org  *****
> > >
> > > Hi everybody,
> > >
> > > The avg. geodesic distance in a (connected) random network is
approximated
> > by ln(n)/ln(k), where n=number_of_nodes , and k=avg._ties_per_node.
However,
> > if I create a random network with parameters n and k in Pajek and
calcluate
> > its avg. geodesic distance, I get sizeably different results (regardless
of
> > how large I set n). Any idea what I could be doing wrong?
> > >
> > >
> > > Thanks,
> > >
> > > Gianluca
> > >
> > >
> > > PS: To calculate the avg. geodesic distance I've been using
> > >
> > >
> > >
> > >
> > >
> > > ____________________________________________________________
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> >
> >
>
> Gianluca Carnabuci
> PhD candidate
> University of Twente
>
> P.O. Box 217
> Enschede
> The Netherlands
>
> Ph: 0031 53 4892352
>
>
>
> ____________________________________________________________
> Libero ADSL: 3 mesi gratis e navighi a 1.2 Mega, senza costi di
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