Euclidean versus hyperbolic congestion in idealized versus experimental networks

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.

💡 Research Summary

This paper provides a rigorous mathematical explanation for the empirically observed phenomenon that, in very large communication networks, traffic tends to concentrate on a very small set of nodes, leading to extreme congestion at those points. The authors argue that this effect arises from the combination of two ingredients: (i) the network’s negative curvature (in the sense of Gromov‑hyperbolicity) and (ii) minimum‑length routing, i.e., packets follow geodesics (shortest paths) between source–destination pairs.

The authors first formalize traffic on a graph G = (V, E) equipped with a symmetric distance function d. For each ordered pair (s, t) a demand Λ_d(s, t) is specified, and a routing protocol selects a shortest‑path geodesic