I found the discussion of the available bandwidth of tree vs higher dimensional virtual networks topologies quite, to quote Spock, fascinating:
A mathematical analysis by Ritter (2002) (one of the original developers
of Napster) presented a detailed numerical argument demonstrating that the
Gnutella network could not scale to the capacity of its competitor, the
Napster network. Essentially, that model showed that the Gnutella network is
severely bandwidth-limited long before the P2P population reaches a million
peers. In each of these previous studies, the conclusions have overlooked the
intrinsic bandwidth limits of the underlying topology in the Gnutella network:
a Cayley tree (Rains and Sloane 1999) (see Sect. 9.4 for the definition).
Trees are known to have lower aggregate bandwidth than higher dimensional
topologies, e.g., hypercubes and hypertori. Studies of interconnection
topologies in the literature have tended to focus on hardware implementations
(see, e.g., Culler et al. 1996; Buyya 1999), which are generally limited
by the cost of the chips and wires to a few thousand nodes. P2P networks,
on the other hand, are intended to support from hundreds of thousands to
millions of simultaneous peers, and since they are implemented in software,
hyper-topologies are relatively unfettered by the economics of hardware.
In this chapter, we analyze the scalability of several alternative topologies
and compare their throughput up to 2–3 million peers. The virtual hypercube
and the virtual hypertorus offer near-linear scalable bandwidth subject to
the number of peer TCP/IP connections that can be simultaneously kept
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