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Geometry of the minimal spanning tree in the heavy-tailed regime: new universality classes

Shankar Bhamidi, Sanchayan Sen

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.

Geometry of the minimal spanning tree in the heavy-tailed regime: new universality classes

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 , the distance between two typical points on the minimal spanning tree (MST) on the giant component in the supercritical regime scales like . 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 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 , 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 . 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.

Paper Structure

This paper contains 20 sections, 33 theorems, 183 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Organization of the paper
  3. Random graph models
  4. Main result
  5. Definitions and preliminary results
  6. Notation
  7. Convergence of metric spaces
  8. $\mathbb{R}$-trees and $\mathbb{R}$-graphs
  9. Some properties of MSTs
  10. Cycle-breaking
  11. A slightly different model
  12. Relating the MST with the components in the critical window
  13. The general strategy
  14. An exploration process
  15. The component of the vertex $1$ in $G_n(\lambda)$
  16. ...and 5 more sections

Key Result

Lemma 2.3

For any $t\geq 0$, the ordered sequence of weights of the connected components of $\mathcal{G}((\mathcal{V}, \boldsymbol{x}), t)$ has the same distribution as the ordered sequence of weights of clusters of $\mathop{\mathrm{MC}}\nolimits((\mathcal{V}, \boldsymbol{x}), t)$.

Figures (2)

  • Figure 2.1: The MST on the component of the vertex $1$ in $\overline\boldsymbol{G}_n$ with $w_i=3(n/i)^{\alpha}$, where $\tau=3.05$ and $n=80000$.
  • Figure 2.2: The MST on the component of the vertex $1$ in $\overline\boldsymbol{G}_n$ with $w_i=3(n/i)^{\alpha}$, where $\tau=3.95$ and $n=80000$.

Theorems & Definitions (46)

  • Definition 2.1: Finite state multiplicative coalescent
  • Definition 2.2: Random graph $\mathcal{G}((\mathcal{V}, \boldsymbol{x}), t)$
  • Lemma 2.3
  • Lemma 2.5
  • Proposition 2.6
  • Definition 2.7: Minimal spanning tree
  • Theorem 3.1
  • Definition 4.1: Real trees legall-surveyevans-book
  • Definition 4.2: $\mathbb{R}$-graphs AddBroGolMie13
  • Lemma 4.3: Minimax paths property
  • ...and 36 more