Hopf bifurcation analysis of a delayed rumor propagation model in microblog networks

Wanmei Hou; Xiangqing Zhao; Wanmei Hou; Xiangqing Zhao

doi:10.3934/nhm.2026065

Networks and Heterogeneous Media

2026, Volume 21, Issue 5: 1739-1766. doi: 10.3934/nhm.2026065

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Hopf bifurcation analysis of a delayed rumor propagation model in microblog networks

Wanmei Hou ¹,
Xiangqing Zhao ^{2
,
,}

1.
School of Marxism, Suqian University, Suqian 223800, China
2.
School of Mathematics and Physics, Suqian University, Suqian 223800, China

Received: 23 May 2026 Revised: 05 July 2026 Accepted: 15 July 2026 Published: 18 August 2026

The rapid proliferation of online rumors undermines social stability, demanding dynamic analyses that transcend traditional steady-state assumptions. This study proposes a delayed Unknowns Neutrals Forwarders Refuters (UNFR) model that introduces discrete time delays into $ R \to U $, $ N \to R $, and $ N \to F $ transitions to account for memory decay and decision-making hesitation. Employing characteristic equation analysis and Hopf bifurcation theory, alongside center manifold and normal form techniques, we examine the system's stability dynamics. We demonstrate that equilibria remain locally asymptotically stable below a critical delay; however, exceeding this threshold induces a Hopf bifurcation, giving rise to sustained periodic oscillations. Numerical simulations corroborate this transition from stability to periodicity, offering a theoretical basis for understanding the cyclical resurgence of rumors and informing effective, time-sensitive intervention strategies.
- online rumors,
- UNFR model,
- time delay,
- Hopf bifurcation,
- cyclical resurgence
Citation: Wanmei Hou, Xiangqing Zhao. Hopf bifurcation analysis of a delayed rumor propagation model in microblog networks[J]. Networks and Heterogeneous Media, 2026, 21(5): 1739-1766. doi: 10.3934/nhm.2026065

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Abstract

The rapid proliferation of online rumors undermines social stability, demanding dynamic analyses that transcend traditional steady-state assumptions. This study proposes a delayed Unknowns Neutrals Forwarders Refuters (UNFR) model that introduces discrete time delays into $ R \to U $, $ N \to R $, and $ N \to F $ transitions to account for memory decay and decision-making hesitation. Employing characteristic equation analysis and Hopf bifurcation theory, alongside center manifold and normal form techniques, we examine the system's stability dynamics. We demonstrate that equilibria remain locally asymptotically stable below a critical delay; however, exceeding this threshold induces a Hopf bifurcation, giving rise to sustained periodic oscillations. Numerical simulations corroborate this transition from stability to periodicity, offering a theoretical basis for understanding the cyclical resurgence of rumors and informing effective, time-sensitive intervention strategies.

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