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Graphons and the $H$-property

Mohamed-Ali Belabbas, Xudong Chen

Abstract

A graphon satisfies the $H$-property if graphs sampled from it contain a Hamiltonian decomposition almost surely, which in turn implies that the corresponding network topologies are, e.g., structurally stable and structurally ensemble controllable. In recent papers, we have exhibited a set of conditions that is essentially necessary and sufficient for the $H$-property to hold for the finite-dimensional class of step-graphons. The extension to the infinite-dimensional case of general graphons was hindered by the fact that said conditions relied on objects that do not admit immediate extensions to the infinite-dimensional case. We outline here our approach to bypass this difficulty and state conditions that guarantee that the $H$-property holds for general graphons.

Graphons and the $H$-property

Abstract

A graphon satisfies the -property if graphs sampled from it contain a Hamiltonian decomposition almost surely, which in turn implies that the corresponding network topologies are, e.g., structurally stable and structurally ensemble controllable. In recent papers, we have exhibited a set of conditions that is essentially necessary and sufficient for the -property to hold for the finite-dimensional class of step-graphons. The extension to the infinite-dimensional case of general graphons was hindered by the fact that said conditions relied on objects that do not admit immediate extensions to the infinite-dimensional case. We outline here our approach to bypass this difficulty and state conditions that guarantee that the -property holds for general graphons.
Paper Structure (3 sections, 5 theorems, 7 equations, 2 figures)

This paper contains 3 sections, 5 theorems, 7 equations, 2 figures.

Table of Contents

  1. Introduction and Problem Formulation
  2. Results for Step-graphons
  3. Extension to General Graphons

Key Result

Theorem 1

For almost all step-graphons $W$, the probability that $\vec{G}_n\sim W$ has a Hamiltonian decomposition tends to either $0$ or $1$. Furthermore,

Figures (2)

  • Figure 1: Left: An undirected graph on $4$ nodes. Right: Its directed counterpart by replacing every undirected edge with two oppositely oriented edges.
  • Figure 2: Left: A step-graphon $W$ with partition $\sigma = (0,0.3,0.6,1)$. Middle: Skeleton graph $S$. Right: Edge polytope $\mathcal{X}(S)$ and the concentration vector $x^*=(0.3,0.3,0.4)$.

Theorems & Definitions (10)

  • Definition 1: $H$-property
  • Definition 2: Step-graphon and its partition
  • Definition 3: Concentration vector
  • Definition 4: Skeleton graph
  • Definition 5: Edge polytope
  • Theorem 1
  • Proposition 2
  • Proposition 3
  • Proposition 4
  • Theorem 5