Stable graphs: distributions and line-breaking construction

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