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

Wanting Gao; Xudong Chen

doi:10.1016/j.ifacol.2025.07.036

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

Wanting Gao
, Xudong Chen

Department of Electrical & Systems Engineering

Research output: Contribution to journal › Conference article › peer-review

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 →. It has been shown in Belabbas and Chen (2023); Belabbas et al. (2021) 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.

Original language	English
Pages (from-to)	7-12
Number of pages	6
Journal	IFAC-PapersOnLine
Volume	59
Issue number	4
DOIs	https://doi.org/10.1016/j.ifacol.2025.07.036
State	Published - Jun 1 2025
Event	10th IFAC Conference on Networked Systems, NECSYS 2025 - Hong Kong, Hong Kong Duration: Jun 2 2025 → Jun 5 2025

Keywords

Graphons
Networked systems
Random graphs
Structural controllability
Structural stability
Structural system theory

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10.1016/j.ifacol.2025.07.036

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title = "On the H-property for Step-graphons: The Residual Case",

abstract = "We investigate the H-property for step-graphons. Specifically, we sample graphs Gn on n nodes from a step-graphon and evaluate the probability that Gn has a Hamiltonian decomposition in the asymptotic regime as n →. It has been shown in Belabbas and Chen (2023); Belabbas et al. (2021) 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.",

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T1 - On the H-property for Step-graphons

T2 - 10th IFAC Conference on Networked Systems, NECSYS 2025

AU - Gao, Wanting

AU - Chen, Xudong

PY - 2025/6/1

Y1 - 2025/6/1

N2 - We investigate the H-property for step-graphons. Specifically, we sample graphs Gn on n nodes from a step-graphon and evaluate the probability that Gn has a Hamiltonian decomposition in the asymptotic regime as n →. It has been shown in Belabbas and Chen (2023); Belabbas et al. (2021) 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.

AB - We investigate the H-property for step-graphons. Specifically, we sample graphs Gn on n nodes from a step-graphon and evaluate the probability that Gn has a Hamiltonian decomposition in the asymptotic regime as n →. It has been shown in Belabbas and Chen (2023); Belabbas et al. (2021) 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.

KW - Graphons

KW - Networked systems

KW - Random graphs

KW - Structural controllability

KW - Structural stability

KW - Structural system theory

UR - https://www.scopus.com/pages/publications/105012459931

U2 - 10.1016/j.ifacol.2025.07.036

DO - 10.1016/j.ifacol.2025.07.036

M3 - Conference article

AN - SCOPUS:105012459931

SN - 2405-8971

VL - 59

SP - 7

EP - 12

JO - IFAC-PapersOnLine

JF - IFAC-PapersOnLine

IS - 4

Y2 - 2 June 2025 through 5 June 2025

ER -

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

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