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##

Small world phenomenon estimated with a fractal argument

`Limewire`, a popular Gnutella client, estimates the network size each day with
a Gnutella Crawler[23]. In the early days of Gnutella, there were 500000
nodes. Now, Gnutella feels competition of other popular file sharing networks
like `eMule` and `Kazaa`; network size dropped down to about 100000 (March 2004).

An interesting problem arises: how many users are reachable by one user in the
Gnutella network? How should one set the count-down timer in each packet to
reach all nodes?
The problem is equivalent to the following one:
how many people are between any two people or between you and the President
of the United Nations, as an example? This is sometimes referred as the small
world problem, a common experience of many of us: we meet a new friend and we
discover he/she knows someone we know as well.
Milgram's experiment showed empirically that any two people in the United
States are distant six edges from each other.

To solve the problem, we attempt a volumetric argument. Some fractal
arguments are given in [26], although they go by far more in depth. However, we start our argument with an analogic machine to compute the solution. Results are consistent with Mandelbrot's
fractal theory [24,25].

**Subsections**

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Tiziano Mengotti
2004-03-27