Abstract: With the exponential growth of internet energy, according to the China Internet Network Information Center (CNNIC), as of June 2021, China’s internet user base stood at 1.011 billion, including 1.007 billion mobile internet users. With the exponential growth of the internet,the spread of online public opinion has an increasing impact on people's lives and social stability. Network-derived public opinion refers to the characteristic of network public opinion evolving into new topics based on the original topic, which will replace the original topic and bring a “secondary impact” to network public opinion events. Currently, studying and analyzing the future trend of network derived public opinion is of great scientific significance not only for guiding and controlling public opinion,but also for maintaining social stability. And it helps government departments at all levels to develop scientific and reasonable public opinion guidance and control measures, and this can further create a healthy and harmonious online public opinion ecological environment.
The dynamic process of online public opinion dissemination can be seen as an uncertain process that changes over time. During the dissemination process, online public opinion is intertwined with rumors, revelations and other information. The active release and forwarding of these messages by netizens can easily cause the original online public opinion to deviate from the evolution law, leading to information alienation, sudden changes in the original online public opinion and a jumping uncertain process, so that the network generates public opinion in this way. Jumping uncertain differential equations are a new mathematical tool for describing dynamic systems in uncertain environments. Based on the analysis and research of potential derivative topics caused by network public opinion, this article fully considers some uncertain factors in the propagation process of network public opinion derivative events. According to the different stages of network public opinion propagation, a corresponding network public opinion propagation model stochastic jump-based uncertain differential equation is proposed.
Finally, based on the actual case of the “Snow Township Car Accident Case”, the actual data on its public opinion development is obtained through the Baidu Index. From the perspective of responding to derivative public opinion,the current focus is mainly on the embryonic and active stages of the development of primary public opinion. During the recession, due to the shift of attention from netizens, the government pays less attention to the derivative public opinion generated by it. However, when the original public opinion during the recession is replaced by new derivative topics, the lack of attention to derivative public opinion may have a worse impact on society. Therefore, the article focuses on exploring the numerical calculation and prediction analysis of the network derived public opinion decline stage model. The forecast outcomes demonstrate a strong alignment with the actual case propagation situation, thereby validating the efficacy of the proposed model.
The network public opinion with derivative public opinion, as a complex evolution form of public opinion development, is intertwined with the original public opinion. The three types of uncertain differential equation models with jumps proposed in this article reflect, to some extent, the propagation laws of network public opinion with derivative public opinion at different stages. The next step of work will attempt to introduce more parameters in these three types of models, making the prediction effect of the model more accurate compared to the actual situation.

Key words: online public opinion, derivative public opinion, differential equations with jump uncertainties

摘要： 带跳不确定微分方程是描述不确定环境下动态系统的新型数学工具,网络舆情传播的动态演化本质上是一种随时间推移呈现显著不确定性的复杂过程。鉴于此,本文以网络衍生舆情的生成机制为切入点,充分考量其传播过程中存在的不确定性特征,依据网络衍生舆情发展的规律,构建了分时期的带跳不确定微分方程模型,以精准刻画网络舆情及其衍生事件的动态传播轨迹。最后结合实际案例“雪乡车祸案”,重点探索了网络衍生舆情衰退模型的数值计算预测分析,预测结果与实际案例的传播态势基本一致,验证了模型的有效性。旨在优化各级政府部门制定科学合理的衍生舆情引导管控方案,为进一步营造健康和谐的网络舆情生态环境提供理论参考。

关键词: 网络舆情, 衍生舆情, 带跳不确定微分方程