Get started
Work

An Investigation into Graph Curvature's Ability to Measure Congestion in Network Flow

Year: 2015
Type: preprint
Abstract: A recent trend in network research involves finding the appropriate geometric model for a given network, to use features of the model to infer information about the network. One piece of information that this paper will focus on is the severity location of congestion in the network's traffic flow. To this end, many versions of surfaces have been proposed, each with some set of parameters to allow the model to fit the network, then those parameters are linked to congestion in some manner. These proposed spaces include Gromov's hyperbolic spaces, a scaled variation of Gromov's hyperbolic space, curved $2$-manifolds, additive metrics, fixed points in the automorphism group of the graph. However, in each case above, the link between the parameters of the geometric space the congestion of the network has been intuitive non-rigorous. This paper is a thorough rigorous treatment of each space's ability to describe congestion. Our conclusion is that Gromov's hyperbolicity is the unique space from which information can be extracted. Our investigation's wide scope led to the resolution of conjectures open problems from a wide variety of topics, including (a) a conjecture of Dourisboure Gavoille on a $2$-approximation method for calculating tree-length, (b) an open problem from Narayan Saniee on the amount of congestion in the Euclidean grid, (c) all of the conjectures from Jonckheere, Lou, Bonahon, Baryshnikov relating congestion to rotational symmetry, and (d) a rejection of the implication by Jonckheere, Lohsoonthorn, Bonahon that scaled hyperbolicity implies properties characterized by hyperbolic spaces. We also show that Buneman's distance approximating tree is the best possible up to a constant both additively multiplicatively.
Language: English
Open Access status: green
OpenAlex ID: W2185579092
Cites: 34
Cited by: 3
Related to: 10