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.Comment: 23 pages, 4 figure

Tevatron Run II combination of the effective leptonic electroweak mixing angle

Drell-Yan lepton pairs produced in the process pp→â.,"+â.,"-+X through an intermediate γ∗/Z boson have an asymmetry in their angular distribution related to the spontaneous symmetry breaking of the electroweak force and the associated mixing of its neutral gauge bosons. The CDF and D0 experiments have measured the effective-leptonic electroweak mixing parameter sin2θefflept using electron and muon pairs selected from the full Tevatron proton-antiproton data sets collected in 2001-2011, corresponding to 9-10 fb-1 of integrated luminosity. The combination of these measurements yields the most precise result from hadron colliders, sin2θefflept=0.23148±0.00033. This result is consistent with, and approaches in precision, the best measurements from electron-positron colliders. The standard model inference of the on-shell electroweak mixing parameter sin2θW, or equivalently the W-boson mass MW, using the zfitter software package yields sin2θW=0.22324±0.00033 or equivalently, MW=80.367±0.017 GeV/c2