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We prove a metric space scaling limit for a critical random graph withindependent and identically distributed degrees having power-law tail behaviourwith exponent $\alpha+1$, where $\alpha \in (1,2)$. The limiting components areconstructed from random $\mathbb{R}$-trees encoded by the excursions above itsrunning infimum of a process whose law is locally absolutely continuous withrespect to that of a spectrally positive $\alpha$-stable L\'evy process. Thesespanning $\mathbb{R}$-trees are measure-changed $\alpha$-stable trees. In eachsuch $\mathbb{R}$-tree, we make a random number of vertex-identifications,whose locations are determined by an auxiliary Poisson process. Thisgeneralises results which were already known in the case where the degreedistribution has a finite third moment (a model which lies in the sameuniversality class as the Erd\H{o}s--R\'enyi random graph) and where the roleof the $\alpha$-stable L\'evy process is played by a Brownian motion.

The stable graph: the metric space scaling limit of a critical random graph with i.i.d. power-law degrees