Rumor propagation dynamics in social networks under the influence of time delay and diffusion

Rumors in social networks are often referred to as infectious diseases of the Internet, because rumors spreading in networks feature strong concealment, fast transmission speed and wide spread. With the development of mobile devices, online rumors nowadays are far more harmful than before. Rumors in social networks show completely different spatiotemporal dynamics from traditional rumor spreading dynamics. A social network rumor spreading model with considering both reaction diffusion and fermentation time delay is proposed in this paper. The effects of spatial diffusion and time delay on rumor spreading in online social networks are studied. Firstly, the existence of equilibrium point of the reaction-diffusion rumor spreading model is analyzed, and the basic regeneration number &lt;inline-formula&gt;&lt;tex-math id="M2"&gt;\begin{document}$R_{0}$\end{document}&lt;/tex-math&gt;&lt;alternatives&gt;&lt;graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="18-20220726_M2.jpg"/&gt;&lt;graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="18-20220726_M2.png"/&gt;&lt;/alternatives&gt;&lt;/inline-formula&gt; is calculated. When &lt;inline-formula&gt;&lt;tex-math id="M3"&gt;\begin{document}$R_{0} &lt; 1$\end{document}&lt;/tex-math&gt;&lt;alternatives&gt;&lt;graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="18-20220726_M3.jpg"/&gt;&lt;graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="18-20220726_M3.png"/&gt;&lt;/alternatives&gt;&lt;/inline-formula&gt;, the rumor stops spreading and disappears in social networks; when &lt;inline-formula&gt;&lt;tex-math id="M4"&gt;\begin{document}$R_{0}&gt;1$\end{document}&lt;/tex-math&gt;&lt;alternatives&gt;&lt;graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="18-20220726_M4.jpg"/&gt;&lt;graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="18-20220726_M4.png"/&gt;&lt;/alternatives&gt;&lt;/inline-formula&gt;, the rumor persists in social networks. Secondly, the local stability of the rumor spreading equilibrium is investigated by using the Roth-Hurwitz stability criterion, and the influence of diffusion on the system stability is discussed. When the diffusion is introduced into a stable rumor spreading model without time delay, the model becomes unstable, indicating that the Turing instability is caused by diffusion. Thirdly, the Hopf bifurcation condition of the rumor spreading model is established by selecting the time delay &lt;i&gt;τ&lt;/i&gt; as the bifurcation parameter, and the expression of bifurcation threshold &lt;inline-formula&gt;&lt;tex-math id="M5"&gt;\begin{document}$\tau_{0}$\end{document}&lt;/tex-math&gt;&lt;alternatives&gt;&lt;graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="18-20220726_M5.jpg"/&gt;&lt;graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="18-20220726_M5.png"/&gt;&lt;/alternatives&gt;&lt;/inline-formula&gt; is given. When &lt;inline-formula&gt;&lt;tex-math id="M6"&gt;\begin{document}$\tau &lt; \tau_{0}$\end{document}&lt;/tex-math&gt;&lt;alternatives&gt;&lt;graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="18-20220726_M6.jpg"/&gt;&lt;graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="18-20220726_M6.png"/&gt;&lt;/alternatives&gt;&lt;/inline-formula&gt;, the rumor propagation model with diffusion term is stable; when &lt;inline-formula&gt;&lt;tex-math id="M7"&gt;\begin{document}$\tau&gt;\tau_{0}$\end{document}&lt;/tex-math&gt;&lt;alternatives&gt;&lt;graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="18-20220726_M7.jpg"/&gt;&lt;graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="18-20220726_M7.png"/&gt;&lt;/alternatives&gt;&lt;/inline-formula&gt;, the model loses the stability and the Hopf bifurcation occurs. The numerical simulation results show that both diffusion and time delay play an important role in the dynamic evolution of rumor spreading. At the same time, the influence of the crowding degree of spreaders on rumor propagation is also simulated. As the crowding gets worse and worse, the rumor refuting effect weakens, the bifurcation threshold &lt;inline-formula&gt;&lt;tex-math id="M8"&gt;\begin{document}$\tau_{0}$\end{document}&lt;/tex-math&gt;&lt;alternatives&gt;&lt;graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="18-20220726_M8.jpg"/&gt;&lt;graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="18-20220726_M8.png"/&gt;&lt;/alternatives&gt;&lt;/inline-formula&gt; decreases, and the propagation peak increases. Therefore, it is important to build an excellent social network environment to supervise the rumors that are still in the fermentation stage, improve the timeliness of the release of rumor refuting information, and strengthen the refuting of rumors among key groups. This paper breaks through the limitation considering only the time evolution, explores the spatiotemporal spreading law of rumor in real society, and provides a new perspective and idea for governing the rumor spreading.