  This paper proposes a mathematical justification of the phenomenon of extreme
congestion at a very limited number of nodes in very large networks. It is
argued that this phenomenon occurs as a combination of the negative curvature
property of the network together with minimum length routing. More
specifically, it is shown that, in a large n-dimensional hyperbolic ball B of
radius R viewed as a roughly similar model of a Gromov hyperbolic network, the
proportion of traffic paths transiting through a small ball near the center is
independent of the radius R whereas, in a Euclidean ball, the same proportion
scales as 1/R^{n-1}. This discrepancy persists for the traffic load, which at
the center of the hyperbolic ball scales as the square of the volume, whereas
the same traffic load scales as the volume to the power (n+1)/n in the
Euclidean ball. This provides a theoretical justification of the experimental
exponent discrepancy observed by Narayan and Saniee between traffic loads in
Gromov-hyperbolic networks from the Rocketfuel data base and synthetic
Euclidean lattice networks. It is further conjectured that for networks that do
not enjoy the obvious symmetry of hyperbolic and Euclidean balls, the point of
maximum traffic is near the center of mass of the network.
