  A well-known open problem on the behavior of optimal paths in random graphs
in the strong disorder regime, formulated by statistical physicists, and
supported by a large amount of numerical evidence over the last decade
[31,32,38,70] is as follows: for a large class of random graph models with
degree exponent $\tau\in (3,4)$, the distance between two typical points on the
minimal spanning tree (MST) on the giant component in the supercritical regime
scales like $n^{(\tau-3)/(\tau-1)}$. The aim of this paper is to make progress
towards a proof of this conjecture.
  We consider a supercritical inhomogeneous random graph model with degree
exponent $\tau\in(3, 4)$ that is closely related to Aldous's multiplicative
coalescent, and show that the MST constructed by assigning i.i.d. continuous
weights to the edges in its giant component, endowed with the tree distance
scaled by $n^{-(\tau-3)/(\tau-1)}$, converges in distribution with respect to
the Gromov-Hausdorff topology to a random compact real tree. Further, almost
surely, every point in this limiting space either has degree one (leaf), or
two, or infinity (hub), both the set of leaves and the set of hubs are dense in
this space, and the Minkowski dimension of this space equals
$(\tau-1)/(\tau-3)$.
  The multiplicative coalescent, in an asymptotic sense, describes the
evolution of the component sizes of various near-critical random graph
processes. We expect the limiting spaces in this paper to be the candidates for
the scaling limit of the MST constructed for a wide array of other heavy-tailed
random graph models.
