  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.
