Title:

Optimal routing in a hyperbolically mapped Internet

Abstract:

We establish a connection between the scale-free topology of complex
networks, and the hyperbolic geometry of hidden metric spaces
underlying these networks. Given a hyperbolic space, networks
topologies with scale-free degree distributions and strong
clustering naturally emerge on top of the space as topological
reflections of its hyperbolic geometry. Conversely, for any
scale-free network with strong clustering, there is an effective
hyperbolic space underlying the network. The underlying hyperbolic
geometry enables greedy routing with optimal efficiency. Greedy
routing does not require any global information about network
topology to navigate the network. At each hop, greedy routing
selects as the next hop the current hop neighbor closest to the
destination in the underlying hyperbolic space. We show that in
complex networks mapped to their hyperbolic spaces, greedy routing
always succeeds reaching the destination, following the
topologically shortest paths. Furthermore, we show that even without
re-mapping the network or changing any node coordinates, this
navigation efficiency is remarkably robust with respect to rapid
network dynamics, and catastrophic levels of network damage. We map
the real Internet AS-level topology to its hyperbolic space, and
find that greedy routing using this map exhibits similar efficiency.
These results effectively deliver a solution for Internet
interdomain routing with theoretically best possible scaling
properties. Not only routing table sizes and stretch are as small as
possible, but also routing communication overhead is reduced to
zero.
