Many applications in network science have recently been discovered for the curvature’’ of a network, but there is no consensus on the definition for this term. A common approach in these applications is to derive from the curvature either a logical center of the network or a tree representation of the network (these would only exist when the curvature is negative), but that such structures can be extracted using curvature alone remains largely conjectural.
A connection between one type of curvature—Gromov’s hyperbolicity—and a tree representation has been known for decades, and recently it has also been connected for unweighted graphs to a logical center.
We extend the connection between Gromov’s hyperbolicity and a logical center to weighted graphs, and we construct counterexamples showing that no other proposed definition for curvature implies the existence of a logical center.
We also consider the leading methods to construct a tree representation of the network and the leading methods to measure the quality of the representation, and show that, despite wildly different descriptions, they are asymptotically equivalent.
These results resolve several conjectures, including a conjecture of Dourisboure and Gavoille on a 2-approximation method for calculating tree-length and all of the conjectures from Jonckheere, Lou, Bonahon, and Baryshnikov relating congestion to rotational symmetry.