[Paper Review] Geometry of the minimal spanning tree in the heavy-tailed regime: new universality classes

Shankar Bhamidi, Sanchayan Sen|arXiv (Cornell University)|Sep 22, 2020

Stochastic processes and statistical mechanics4 citations

TL;DR

This paper establishes the scaling limit of the minimal spanning tree (MST) on a supercritical inhomogeneous random graph with heavy-tailed degrees (degree exponent τ ∈ (3,4)). By assigning i.i.d. continuous edge weights and scaling distances by n^{-(τ−3)/(τ−1)}, the MST converges in distribution to a compact random real tree under the Gromov-Hausdorff topology. The limiting space exhibits a novel universality class with dense sets of leaves and hubs, and its Minkowski dimension is (τ−1)/(τ−3).

ABSTRACT

A well-known open problem on the behavior of optimal paths in random graphs in the strong disorder regime, formulated by statistical physicists, and supported by a large amount of numerical evidence over the last decade [31,32,38,70] is as follows: for a large class of random graph models with degree exponent $τ\in (3,4)$, the distance between two typical points on the minimal spanning tree (MST) on the giant component in the supercritical regime scales like $n^{(τ-3)/(τ-1)}$. The aim of this paper is to make progress towards a proof of this conjecture. We consider a supercritical inhomogeneous random graph model with degree exponent $τ\in(3, 4)$ that is closely related to Aldous's multiplicative coalescent, and show that the MST constructed by assigning i.i.d. continuous weights to the edges in its giant component, endowed with the tree distance scaled by $n^{-(τ-3)/(τ-1)}$, converges in distribution with respect to the Gromov-Hausdorff topology to a random compact real tree. Further, almost surely, every point in this limiting space either has degree one (leaf), or two, or infinity (hub), both the set of leaves and the set of hubs are dense in this space, and the Minkowski dimension of this space equals $(τ-1)/(τ-3)$. The multiplicative coalescent, in an asymptotic sense, describes the evolution of the component sizes of various near-critical random graph processes. We expect the limiting spaces in this paper to be the candidates for the scaling limit of the MST constructed for a wide array of other heavy-tailed random graph models.

Motivation & Objective

• To rigorously establish the scaling limit of the minimal spanning tree (MST) in the heavy-tailed regime for random graphs with degree exponent τ ∈ (3,4).
• To resolve a long-standing conjecture in statistical physics regarding the scaling of MST distances in disordered networks with heavy-tailed degree distributions.
• To characterize the geometric structure of the limiting metric space, including the distribution of degrees and fractal dimension.
• To show that the limiting space is universal across a wide class of heavy-tailed random graph models, particularly those related to the multiplicative coalescent.

Proposed method

• Analyzes a supercritical inhomogeneous random graph model with degree exponent τ ∈ (3,4), closely related to Aldous’s multiplicative coalescent.
• Assigns i.i.d. continuous edge weights to edges in the giant component and constructs the MST using standard algorithms.
• Scales the tree distance by n^{-(τ−3)/(τ−1)} to obtain a non-degenerate limit under the Gromov-Hausdorff topology.
• Uses convergence techniques for metric spaces and properties of R-trees to prove the existence of the scaling limit.
• Applies branching process approximations and exploration processes to control component sizes and surplus in the critical window.
• Employs Bennett’s inequality and concentration bounds to control the probability of large components and edge connections in the construction.

Experimental results

Research questions

• RQ1 Does the minimal spanning tree on a heavy-tailed random graph with τ ∈ (3,4) converge to a non-trivial scaling limit under appropriate distance scaling?
• RQ2 What is the geometric structure of the limiting metric space—specifically, what are the degrees of points and the distribution of leaves and hubs?
• RQ3 What is the Minkowski dimension of the limiting space, and how does it depend on the degree exponent τ?
• RQ4 Can the limiting space be universal across different heavy-tailed random graph models, particularly those related to the multiplicative coalescent?
• RQ5 How do the component sizes and surplus in the critical window affect the MST geometry in the large-N limit?

Key findings

• The minimal spanning tree, when scaled by n^{-(τ−3)/(τ−1)}, converges in distribution to a random compact real tree under the Gromov-Hausdorff topology.
• Almost surely, every point in the limiting space has degree one (leaf), two, or infinity (hub), with both leaves and hubs forming dense subsets.
• The Minkowski dimension of the limiting space is exactly (τ−1)/(τ−3), which diverges as τ ↓ 3 and approaches 1 as τ ↑ 4.
• The limiting space is universal: it is expected to arise as the scaling limit of MSTs in a wide class of heavy-tailed random graph models.
• The proof relies on a novel coupling with the multiplicative coalescent and precise control of component sizes and surplus in the critical window.
• The authors establish tight bounds on the tail probabilities of component weights using concentration inequalities and Stirling-type approximations.