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On the $H$-property for Step-graphons: The Residual Case

Wanting Gao, Xudong Chen

TL;DR

This work analyzes the H-property for step-graphons in the residual case, where the probability that a sampled graph G_n has a Hamiltonian decomposition does not converge to 0 or 1. It proves that the limit exists and provides an explicit expression: it equals the probability that a Gaussian vector ω^* with mean x^* and covariance Σ lies in a polyhedral region Ω(x^*), defined by the facet hyperplanes of the edge cone associated with the skeleton graph S. The result complements the established zero-one theorem by fully characterizing the remaining scenario and is validated through detailed numerical experiments on three step-graphons, showing agreement with the predicted Gaussian limit. The findings deepen understanding of how network structure and concentration heterogeneity govern ensemble controllability properties in graphon-generated graphs.

Abstract

We investigate the $H$-property for step-graphons. Specifically, we sample graphs $G_n$ on $n$ nodes from a step-graphon and evaluate the probability that $G_n$ has a Hamiltonian decomposition in the asymptotic regime as $n\to\infty$. It has been shown that for almost all step-graphons, this probability converges to either zero or one. We focus in this paper on the residual case where the zero-one law does not apply. We show that the limit of the probability still exists and provide an explicit expression of it. We present a complete proof of the result and validate it through numerical studies.

On the $H$-property for Step-graphons: The Residual Case

TL;DR

This work analyzes the H-property for step-graphons in the residual case, where the probability that a sampled graph G_n has a Hamiltonian decomposition does not converge to 0 or 1. It proves that the limit exists and provides an explicit expression: it equals the probability that a Gaussian vector ω^* with mean x^* and covariance Σ lies in a polyhedral region Ω(x^*), defined by the facet hyperplanes of the edge cone associated with the skeleton graph S. The result complements the established zero-one theorem by fully characterizing the remaining scenario and is validated through detailed numerical experiments on three step-graphons, showing agreement with the predicted Gaussian limit. The findings deepen understanding of how network structure and concentration heterogeneity govern ensemble controllability properties in graphon-generated graphs.

Abstract

We investigate the -property for step-graphons. Specifically, we sample graphs on nodes from a step-graphon and evaluate the probability that has a Hamiltonian decomposition in the asymptotic regime as . It has been shown that for almost all step-graphons, this probability converges to either zero or one. We focus in this paper on the residual case where the zero-one law does not apply. We show that the limit of the probability still exists and provide an explicit expression of it. We present a complete proof of the result and validate it through numerical studies.

Paper Structure

This paper contains 10 sections, 9 theorems, 55 equations, 8 figures.

Table of Contents

  1. Introduction
  2. The $H$-property
  3. Step-graphons and the almost zero-one property
  4. Main Result for the Residual Case
  5. Proof of Theorem \ref{['thm:main']}
  6. Numerical Study
  7. Example 1
  8. Example 2
  9. Example 3
  10. Conclusion

Key Result

Lemma 1

Let $S$ be the skeleton graph of $W$ on $q$ nodes. Suppose that $S$ is connected; then,

Figures (8)

  • Figure 1: (a) A step-graphon $W$ with partition $\sigma = (0, 0.25, 0.5, 0.75, 1)$, where the value of the graphon is coded by shade, with white being 0 and black being 1. (b) The associated skeleton graph $S$. (c) An undirected graph $G$ sampled from $W$. (d) The directed graph $\vec{G}$ obtained from $G$, with cycles $D_1 = v_1v_2v_4v_3v_1$ and $D_2 = v_5v_6v_5$ forming a Hamiltonian decomposition of $\vec{G}$.
  • Figure 2: Three step-graphons shown in (a), (b), and (c) with the same skeleton graph, but different concentration vectors. The shaded region in (d) is the edge polytope $\mathcal{X}(S)$, embedded in the standard simplex $\Delta^{2}$. The red, green, and orange dots are the concentration vectors $x^*$ for the above three step-graphons. We have that (a) $x^* \in \operatorname{int} \mathcal{X}(S)$ and $\lim_{n\to\infty}\mathbf{P}(\mathcal{E}_n) = 1$; (b) $x^* \in \partial(\mathcal{X}(S))$ and $\lim_{n\to\infty}\mathbf{P}(\mathcal{E}_n) = 0.5$; and (c) $x^* \notin \mathcal{X}(S)$ and $\lim_{n\to\infty}\mathbf{P}(\mathcal{E}_n) = 0$.
  • Figure 3: $(a)$ The step-graphon $W_1$. $(b)$ The associated edge polytope $\mathcal{X}(S)$ with concentration vector $x^*$.
  • Figure 4: Empirical probability that ${\vec{G}_n}\sim W_1$ has an HD.
  • Figure 5: Skeleton graph associated with $W_2$ and $W_3$
  • ...and 3 more figures

Theorems & Definitions (16)

  • Definition 1: $H$-property
  • Definition 2: Step-graphon and its partition
  • Definition 3: Concentration vector
  • Definition 4: Skeleton graph
  • Definition 5: Incidence matrix
  • Definition 6: Edge polytope
  • Lemma 1
  • Theorem 1
  • Lemma 2
  • Definition 7
  • ...and 6 more