Rumor spreading in complex networks under stochastic node activity

Abstract

The research on rumor spreading has a long history, and its wanton flooding has brought huge impact on people's life. In the process of its spreading, the individual's activity plays an important role. However, in the complex and changeable environment, randomness cannot be ignored, not to mention its influence on individual activity Based on the $ISR$ model of individual activity, this paper explores the stochastic version of the rumor model including fluctuations in the activity. Then, the influence of Stratonovich stochastic noise on the asymptotic behavior of non-linear rumor spreading model is studied. Through the mathematical analysis, we get the critical values to measure whether the deterministic and stochastics models spread or not, as well as the threshold conditions for rumor to spread wantonly. At the same time, the effects of Stratonovich stochastic noise on the asymptotic behavior of rumor-free equilibrium point $E^0$ and endemic equilibrium point $E^{\ast }$ are obtained respectively, and the condition that the rumor-free equilibrium is globally asymptotically stable in the presence of noise is given. Finally, the theoretical results are verified by numerical simulation.

Keywords: Rumor spreading; Social network dynamics; Stochastic differential equation; Wiener process.