| The concept of curvature of communication networks is investigated through the theory of δ-hyperbolic space, which can be intuitively viewed as the generalization of Riemannian manifolds with negative curvature to metric graphs. The hyperbolic measure δ can be expressed in terms of slimness, insize, thinness, and fatness of geodesic triangles in the metric space. The analytical formula for the slimness, insize, thinness, and fatness are computed in terms of the internal angles of the geodesic triangles and the curvature κ of the underlying Riemannian manifold with constant negative curvature κ. In addition, the fatness of a geodesic triangle with acute angles only can be given a billiard dynamics interpretation, in the sense that the optimum inscribed triangle is the period three orbit of the billiard dynamics on a geodesic triangular table.; To assess the hyperbolic property of communication networks, the mathematical expectation of δ over the diameter for several random graph generators is computed by Monte Carlo simulation. Among random graphs, small world graphs, and scale free generators, the scale free model, which is used as a topology generator in communication network, appears to be the most hyperbolic. This result is an extra piece of evidence of the hyperbolic property of the internet which has already been claimed by two different groups, using other arguments, though.; With the evidence of the δ-hyperbolic property of the internet, multi-path routing can be achieved along quasi-geodesics, which can be computed via k-local geodesic paths. It turns out that the alternative paths are sufficiently close to the optimum path. To assess the closeness between geodesic and quasi-geodesics, an upper bound on the Hausdorff distance between the geodesic and quasi-geodesics is derived for Riemannian manifolds with constant negative curvature and general δ-hyperbolic geodesic spaces. |
