On Tuesday, March 21, 2006 Oskar Sandberg wrote:
> It could probably grow forever without having a phase transition.
I sure hope so, but then the next question is, what should be
the scaling factor k in "degree=k x log2(N)" for this to be the case?
I mean, k=1 seems to be not enough - with k=1 you have a transition
with 1M nodes (when your degree is 20), correct?
So what is the needed k? Does the theory say anything about it?
And more importantly, when you deviate from the ideal (random-pairs)
rewiring algorithm due to the caching present in the system, what
should be your practical value of k to give you a safety margin
sufficient to avoid the phase transition at any network size?
In practice, of course, people tend to work with k>>1, if only
to minimize the number of hops and to increase the network robustness
(like this Kad example with hundreds of links). But still it might be
interesting to see whether any value of k used in practice would be
theoretically enough to handle any practically possible content
popularity distributions, caching, and such.
What do you think about all this?
Best wishes -
S.Osokine,
Crackpot Physicist.
21 Mar 2006.
-----Original Message-----
From: [EMAIL PROTECTED] [mailto:[EMAIL PROTECTED]
Behalf Of Oskar Sandberg
Sent: Tuesday, March 21, 2006 1:00 PM
To: Peer-to-peer development.
Subject: Re: Big hype on small worlds. (was Re: Dijjerand Freenet(RE:
[p2p-hackers] clustering))
Serguei Osokine wrote:
> On Tuesday, March 21, 2006
>
>>I reran the same simulations I posted before, this time scaling
>>up the degrees with the size of the network (2 log_2 N edges per
>>node)...
>
>
> Out of curiosity, what happens to the phase transition with
> this number of degrees? 1M meltdown was with 20 links, right? Does
> this extra 2x avoid the phase transition at 1M nodes, and if so,
> does it merely push it to say, 2M, or it is completely avoided
> for all practical node counts (say, less than 10^12 or something)?
When scaling the node degree with log N I have never observed the number
of successful searches decreasing with the network size. In this
particular case there was not a single failed search for any network
size. There are (slightly less than rigorous) mathematical reasons why
this should be the case. It could probably grow forever without having
a phase transition.
// oskar
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