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; we will observe that these distributions are related to the distributions of some configuration models (2) determine the distribution of the α-stable graph 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.Figure 1: A simulation of a connected component of the stable graph when α = 1.5 and the surplus is 2. The cycle structure is shown in black.

We construct random metric spaces by gluing together an infinite sequence of pointed metric spaces that we call blocks. At each step, we glue the next block to the structure constructed so far by randomly choosing a point on the structure and then identifying it with the distinguished point of the block. The random object that we study is the completion of the structure that we obtain after an infinite number of steps. In [7], Curien and Haas study the case of segments, where the sequence of lengths is deterministic and typically behaves like n −α . They proved that for α > 0, the resulting tree is compact and that the Hausdorff dimension of its set of leaves is α −1 . The aim of this paper is to handle a much more general case in which the blocks are i.i.d. copies of the same random metric space, scaled by deterministic factors that we call (λ n ) n≥1 . We work under some conditions on the distribution of the blocks ensuring that their Hausdorff dimension is almost surely d, for some d ≥ 0. We also introduce a sequence (w n ) n≥1 that we call the weights of the blocks. At each step, the probability that the next block is glued onto any of the preceding blocks is proportional to its weight. The main contribution of this paper is the computation of the Hausdorff dimension of the set L of points which appear during the completion procedure when the sequences (λ n ) n≥1 and (w n ) n≥1 typically behave like a power of n, say n −α for the scaling factors and n −β for the weights, with α > 0 and β ∈ R. For a large domain of α and β we have the same behaviour as the one observed in [7], which is that dim H (L) = α −1 . However for β > 1 and α < 1/d, our results reveal an interesting phenomenon: the dimension has a non-trivial dependence in α, β and d, namelyThe computation of the dimension in the latter case involves new tools, which are specific to our model. arXiv:1707.09833v2 [math.PR]

We study two models of growing recursive trees. For both models, the tree initially contains a single vertex u1 and at each time n ≥ 2 a new vertex un is added to the tree and its parent is chosen randomly according to some rule. In the weighted recursive tree, we choose the parent u k of un among {u1, u2, . . . , un−1} with probability proportional to w k , where (wn) n≥1 is some deterministic sequence that we fix beforehand.In the affine preferential attachment tree with fitnesses, the probability of choosing any u k is proportional to a k + deg + (u k ), where deg + (u k ) denotes its current number of children, and the sequence of fitnesses (an) n≥1 is deterministic and chosen as a parameter of the model. We show that for any sequence (an) n≥1 , the corresponding preferential attachment tree has the same distribution as some weighted recursive tree with a random sequence of weights (with some explicit distribution). We then prove almost sure scaling limit convergences for some statistics associated with weighted recursive trees as time goes to infinity, such as degree sequence, height, profile and also the weak convergence of some measures carried on the tree. Thanks to the connection between the two models, these results also apply to affine preferential attachment trees.

The aim of this paper is to develop a method for proving almost sure convergence in the Gromov-Hausdorff-Prokhorov topology for a class of models of growing random graphs that generalises Rémy's algorithm for binary trees. We describe the obtained limits using some iterative gluing construction that generalises the famous line-breaking construction of Aldous' Brownian tree, and we characterize some of them using the self-similarity property in law that they satisfy.To do that, we develop a framework in which a metric space is constructed by gluing smaller metric spaces, called blocks, along the structure of a (possibly infinite) discrete tree. Our growing random graphs seen as metric spaces can be understood in this framework, that is, as evolving blocks glued along a growing discrete tree structure. Their scaling limit convergence can then be obtained by separately proving the almost sure convergence of every block and verifying some relative compactness property for the whole structure. For the particular models that we study, the discrete tree structure behind the construction has the distribution of an affine preferential attachment tree or a weighted recursive tree. We strongly rely on results concerning those two models and their connection, obtained in the companion paper Sénizergues (2021).

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