Stable graphs: distributions and line-breaking construction

For α ∈ ( 1 , 2 ] , the α -stable graph arises as the universal scaling limit of critical random graphs with i.i.d. degrees having a given α -dependent power-law tail behavior. It consists of a sequence of compact measured metric spaces (the limiting connected components), each of which is tree-like, in the sense that it consists of an ℝ -tree with finitely many vertex-identifications (which create cycles). Indeed, given their masses and numbers of vertex-identifications, these components are independent and may be constructed from a spanning ℝ -tree, which is a biased version of the α -stable tree, with a certain number of leaves glued along their paths to the root. In this paper we investigate the geometric properties of such a component with given mass and number of vertex-identifications. We (1) obtain the distribution of its kernel and more generally of its discrete finite-dimensional marginals, and observe that these distributions are themselves related to the distributions of certain configuration models; (2) determine its distribution as a collection of α -stable trees glued onto its kernel; and (3) present a line-breaking construction, in the same spirit as Aldous’ line-breaking construction of the Brownian continuum random tree.