Applying the formula

A homogeneous Gnutella is a fractal with dimension $d=7.79$ according to the following table that lists some related problems estimated with our formula, in bold text we highlight the computed result, the other numbers are assumptions:

Description	$N$	$M$	$d$	$l$
Gnutella	$500000$	$\mathbf{15.58}$	$\mathbf{7.79}$	$7$ (TTL)
Milgram's experiment	$250 \cdot 10^{6}$	$\mathbf{24.38}$	$\mathbf{12.19}$	$6$
World	$6 \cdot 10^{9}$	$24.38$	$12.19$	$\mathbf{7.78}$
Switzerland	$7 \cdot 10^{6}$	$24.38$	$12.19$	$\mathbf{4.47}$
Poschiavo	$3500$	$24.38$	$12.19$	$\mathbf{2.39}$
CPU	$3 \cdot 10^{6}$	$3$	$1.5$	$\mathbf{27257}$
Brain	$100 \cdot 10^{9}$	$10$	$5$	$\mathbf{218.67}$

For Gnutella, we know $l$ as the standard TTL⁸ and the number of nodes computed by the Gnutella crawler[23]. Using the formula, we compute the dimension $d$ and the average number of outgoing connections $M$.

Milgram's experiment showed that in the United States, a country with about 250 million ($N$) inhabitants, there are 6 ($l$) degrees of separation. The formula estimates that each person knows about 25 people enough well to perform the experiment.

Using Milgram's $M$, we compute $l$ for the entire world, for Switzerland and for a little village in the mountains.

We try the formula on the brain, a complex network with 100 billion neurons. Each neuron has about 15000 connections but is connected to a neighborhood of about 10 other neurons only, about 1500 connections for each neuron. The path length would be then 218. We idealized the CPU as it would be composed by 3 million NAND ports, with 2 inputs and one output.

However, Gnutella and the above mentioned problems are far from homogeneous in the degree of the nodes, so that our formula gives only a rough estimation.