Euclidean versus hyperbolic congestion in idealized versus experimental networks

TL;DR

This paper provides a mathematical explanation for the extreme congestion at certain nodes in large networks, highlighting differences between hyperbolic and Euclidean geometries and supporting experimental observations.

Contribution

It offers a theoretical comparison of traffic congestion in hyperbolic versus Euclidean networks, explaining observed discrepancies in traffic load scaling.

Findings

Hyperbolic networks concentrate traffic at the center regardless of size.

Euclidean networks show traffic concentration that diminishes with network size.

Hyperbolic networks exhibit quadratic scaling of traffic load at the center.

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 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.