Is it possible to treat large scale distributed systems as physical
systems? The importance of that question stems
from the fact that the
behavior of many P2P systems is very complex to analyze analytically,
and simulation of scales of interest can be prohibitive. In Physics,
however, one is accustomed to reasoning about large systems. The limit
of very large systems may actually simplify the analysis. As a first
step, we here analyze the effect of the density of
populated nodes in
an identifier space in a P2P system. We show that while the average
path length is approximately given by a function of the number of
populated nodes, there is a systematic
correction which depends on the
density. In other words, the dependence is both on the number of
address nodes and the number of populated nodes, but only through their
ratio. Interestingly, the correction is negative for finite densities,
showing that an amount of randomness somewhat shortens average path
length. END of example.