Abstract. 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 Θ(1), whereas in a Euclidean ball, the same proportion scales as Θ (1/R n −1 ). This discrepancy persists for the traffic load, which at the center of the hyperbolic ball scales as volume 2 (B), whereas the same traffic load scales as volume 1+ 1/ n (B) 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 database 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.
Abstract-The technique of effective resistance has seen growing popularity in problems ranging from escape probability of random walks on graphs to asymptotic space localization in sensor networks. The results obtained thus far deal with such problems on Euclidean lattices, on which their asymptotic nature already reveals that the crucial issue is the large scale behavior of such lattices. Here we investigate how such results have to be amended on a class of graphs, referred to as Gromov hyperbolic, which behave in the large scale as negatively curved Riemannian manifolds. It is argued that Gromov hyperbolic graphs occur quite naturally in many situations. Among the results developed here, we will mention the nonvanishing probability of escape of a random walk to a Cantor set Gromov boundary and the facts that the space localization error of sensors networked in a Gromov hyperbolic fashion grows linearly with the distance to a sensor whose geographical position is known, but would become uniformly bounded in an idealized situation in which the geographical locations of the nodes at the Gromov boundary are known
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