Convergence rate of $H$-property for step-graphons

Abstract: A graphon is said to have the $H$-property if a random undirected graph $G_n$ on $n$ nodes sampled from it has a node-wise disjoint cycle cover almost surely as $n\to\infty$. It has been shown in the earlier work that the $H$-property obeys the zero-one law, i.e., the probability that the random graph has a cycle cover tends to either one or zero. In this paper, we sharpen the result by characterizing the convergence rate of the probability. Specifically, we show that there are two different types of rates, with one being exponential and the other being root $n$. We provide a rigorous proof and numerical validation.

• The paper presents a precise quantification of convergence rates for the H-property in step-graphons by distinguishing exponential decay from root-n convergence.
• It employs advanced probabilistic techniques and geometric analysis of the skeleton graph, edge cone, and concentration vectors to derive convergence criteria.
• Empirical validation and theoretical insights offer practical guidance for network design and robust system analysis in random graph models.

Convergence Rate Analysis of the $H$-Property in Step-Graphons

Introduction

This work presents a precise characterization of the convergence rate for the $H$-property in step-graphons, building directly on established zero-one laws for random graphs generated from graphons. The $H$-property centers on the occurrence of node-wise disjoint cycle covers in graphs sampled from a given graphon. Prior research established that, for most step-graphons, the probability $P_n(W)$ (that a random graph $G_n \sim W$ on $n$ vertices possesses a cycle cover) converges to either 0 or 1, except for a residual set of cases [belabbas2021h, belabbas2023geometric, gao2025h]. This paper’s contribution lies in quantifying the speed of this convergence and separating regimes in which the rate is exponential from those governed by a root-$n$ decay.

Theoretical Framework

A step-graphon $W$ is defined via a partition $\sigma$ of $[0,1]$, with $H$0 constant on each rectangle of the form $H$1. The probabilistic structure of $H$2 is described by the "concentration vector" $H$3 (probabilities associated with steps) and the "skeleton graph" $H$4 (encoding the support of $H$5 in terms of adjacencies among the steps). The "edge cone" $H$6 is the positive hull generated by incidence vectors of edges in $H$7.

The pivotal conditions for convergence rates are as follows:

• Condition A: Skeleton $H$8 has an odd cycle.
• Condition B: $H$9 lies in the relative interior of the edge cone $H$0.
• Condition B′: $H$1 belongs to $H$2.

These geometric objects organize the zero-one law and, as this paper shows, control finer convergence rates.

Main Results

The main theorem provides four precise assertions delineating the possible convergence rates and their necessary and sufficient geometric criteria:

1. Exponential convergence to one: If $H$3 has an odd cycle and $H$4, then $H$5 exponentially fast, i.e., $H$6 for some $H$7.
2. Exponential convergence to zero: If $H$8, then $H$9 exponentially; if $P_n(W)$0 has an odd cycle but $P_n(W)$1 lies on the boundary of $P_n(W)$2 and $P_n(W)$3 is degenerate, the rate can also be exponential.
3. Root-$P_n(W)$4 convergence to zero: If $P_n(W)$5 (the boundary), but $P_n(W)$6 has no odd cycle, then $P_n(W)$7.
4. Root-$P_n(W)$8 convergence to a nontrivial limit: If $P_n(W)$9 has an odd cycle and $G_n \sim W$0 but not in the interior, then $G_n \sim W$1 converges to $G_n \sim W$2 at rate $G_n \sim W$3.

The explicit form of $G_n \sim W$4 for the residual case is connected to Gaussian limits on affine slices of $G_n \sim W$5 determined by the local geometry of the edge cone near $G_n \sim W$6.

Proof Techniques

Several advanced methods from extremal combinatorics, probabilistic combinatorics, and high-dimensional probability are deployed:

• Decomposition into key geometric objects: The skeleton graph $G_n \sim W$7 and the edge cone $G_n \sim W$8 transform the combinatorial structure of the cycle cover problem into geometric constraints on empirical concentrations $G_n \sim W$9.
• Regularity and embedding via the Blow-up Lemma: The embedding of large cycle covers is facilitated provided that $n$0 is $n$1-super-regular in the sense induced by $n$2, an event that holds with high probability if $n$3 is close to $n$4, and $n$5 lies deeply inside $n$6 [komlos1997blow].
• Concentration inequalities and Berry–Esseen bounds: The distribution of $n$7 is controlled by Hoeffding’s inequality and Berry–Esseen-type results for multinomial random vectors, yielding both exponential and root-$n$8 deviation controls, depending on whether $n$9 lies inside $n$0 or on the boundary [hoeffding1963probability, bentkus2003dependence].
• Spectral-geometric analysis: The dimension of $n$1 (determined via the presence of odd cycles in $n$2) is essential for distinguishing between sharp and degenerate boundary phenomena.

Numerical Analysis

Comprehensive Monte Carlo validation confirms both the exponential and root-$n$3 regimes. Varying the position of $n$4 within or approaching the boundary of $n$5 results in strong modulation of the observed convergence rate. In the exponential regime, explicit rates decrease as $n$6 approaches $n$7. In the root-$n$8 regime, log-log plots yield slopes commensurate with the predicted $n$9, corroborating the theoretical predictions.

Implications and Outlook

These results provide fine-grained asymptotic control for a broad class of random graph models with structured inhomogeneity (step-graphons), precisely characterizing the risk profile for the emergence of cycle covers—a key property in networked control and robust system design.

The techniques developed offer directions for the analysis of other structural properties in graphon-based models, with potential extensions to more general classes of geometric graphons and for other global topological structures (e.g., Hamiltonicity or $W$0-factors). The geometric probability lens (via edge cones and concentration phenomena) is likely adaptable beyond the classical step-graphon regime.

For applications, this analysis enables practitioners to compute or bound the rate at which large random networks drawn from a prescribed blueprint approach desirable connectivity or controllability thresholds, giving explicit guidance on scaling behavior and critical window phenomena.

Conclusion

This paper establishes a sharp dichotomy in the convergence rates for the $W$1-property in step-graphons, controlling for the local position of the empirical concentration vector relative to the edge cone associated with the skeleton graph. The main theorem positions the exponential versus root-$W$2 regime as a function of combinatorial-geometric criteria, unifying previously qualitative laws into precise quantitative statements and providing both proof and empirical substantiation of the claims. These findings further refine the probabilistic understanding of random combinatorial structures generated by graphon models and open the route to generalized structural and extremal questions in inhomogeneous random graph settings.

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Key references:

• "On the H-property for step-graphons and edge polytopes" [belabbas2021h]
• "Geometric Characterization of the H-property for Step-graphons" [belabbas2023geometric]
• "On the H-property for Step-graphons: The Residual Case" [gao2025h]
• Berry–Esseen bounds for high-dimensional probability [bentkus2003dependence]
• Blow-up Lemma and extremal embedding [komlos1997blow]
• Convergence rate analysis for multinomial distributions [arenbaev1977asymptotic, hoeffding1963probability, vershynin2018high]