Stable graphs: distributions and line-breaking construction

Abstract

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

The Critical Beta-splitting Random Tree II: Overview and Open Problems

“…Such a construction is consistent as k increases, and exchangeable over the random leaves, and indeed this constitutes one general approach to the construction of continuum random trees (CRTs) [3,14]. Our inductive construction in section 3 is analogous to the line-breaking construction of the Brownian CRT [2] and stable trees [16].…”

“…This is strongly supported by numerical calculations. In fact, inspired by (16), one might conjecture a stronger relation…”

Stable Densities, Fractional Integrals and the Mittag-Leffler Function

“…3-parameter Mittag-Leffler Distribution: ML(α, β, γ) ≡ ML(α, β, γ, 0) [5,12,19,40,58] 3. Beta-Mittag-Leffler Distribution: BML(α, θ, β) ≡ ML(α, β, 0, θ > 0) [2,12,40,16] 4. 2-parameter/Generalized Mittag-Leffler Distribution: ML(α, θ) ≡ ML(α, 1, 1, θ > −α), equivalently ML(α, 0, 0, θ > 0) [2,8,16,17,22,23,24,28,45] 5.…”

“…In another context, Mittag-Leffler distributions play a fundamental role in the probabilistic modelling of the evolution of random trees and graphs, often used as models of real-world network behaviour. In the study of stable trees (Goldschmidt and Haas [17], Rembart and Winkel [49,50]) and stable graphs (Goldschmidt et al [16]), the distributions that feature as building blocks are beta, generalised Mittag-Leffler ML(α, θ), Dirichlet and Poisson-Dirichlet PD(α, θ). James [24], Sénizergues [55] discussed the appearance of ML(α, θ) as limit distributions in the growth of random graphs inspired by the preferential attachment model due to Barabási and Albert [3].…”

“…The particular case β = γ = 1 reproduces ML(α, θ) ≡ ML(α, 1, 1, θ). This defines the Mittag-Leffler Markov Chain (MLMC) with recursion of the form T n−1 = T n U n ( [16,17,22,49,50,55]). MLMC features in models of the growth of random trees.…”