An improved ISR-WV rumor propagation model based on multichannels with time delay and pulse vaccination

The rapid development of the Internet has broadened the channels of dissemination of information,it has also led to the rapid and widespread propagation of rumors,which can have a serious negative impact socially.In this paper,an improved ISR-WV rumor propagation model integrating multichannels is proposed by considering the system’s time delay,and the influence of different channels of propagation on the dynamic process is further analyzed.Moreover,the basic reproduction number R0,rumor-free equilibrium,and rumor-prevailing equilibrium,as well as their stability,are deduced.Then,an optimal control problem with pulse vaccination is designed.Finally,the validity of the model and theoretical results is verified by numerical simulations and a practical application.The results show that the rumor propagation threshold R0 is more sensitive to the rate of the propagation of the information base channel.The shorter the thinking timeτ_(1)required for the ignorant to react after obtaining the information,the larger the final scale of propagation.Under this condition,the time delayτ_(2)spent by a spreader in producing a video is negatively related to the final scale of the propagation;conversely,a longerτ_(1)implies that the person tends to more cognizant,which can suppress the spread of rumors.Under this condition,τ_(2)has little effect on the final scale of propagation.In addition,the results also prove that timely implementation of the pulse vaccination control strategy of popular science education can effectively control the propagation of rumors and reduce their negative impact.

Correlation and trust mechanism-based rumor propagation model in complex social networks

In real life, the rumor propagation is influenced by many factors. The complexity and uncertainty of human psychology make the diffusion model more challenging to depict. In order to establish a comprehensive propagation model, in this paper, we take some psychological factors into consideration to mirror rumor propagation. Firstly, we use the Ridenour model to combine the trust mechanism with the correlation mechanism and propose a modified rumor propagation model. Secondly, the mean-field equations which describe the dynamics of the modified SIR model on homogenous and heterogeneous networks are derived. Thirdly, a steady-state analysis is conducted for the spreading threshold and the final rumor size. Fourthly, we investigate rumor immunization strategies and obtain immunization thresholds. Next, simulations on different networks are carried out to verify the theoretical results and the effectiveness of the immunization strategies.The results indicate that the utilization of trust and correlation mechanisms leads to a larger final rumor size and a smaller terminal time. Moreover, different immunization strategies have disparate effectiveness in rumor propagation.

Dynamics of a stochastic rumor propagation model incorporating media coverage and driven by Lévy noise

With the development of information technology,rumors propagate faster and more widely than in the past.In this paper,a stochastic rumor propagation model incorporating media coverage and driven by Lévy noise is proposed.The global positivity of the solution process is proved,and further the basic reproductive number R_(0) is obtained.When R_(0)<1,the dynamical process of system with Lévy jump tends to the rumor-free equilibrium point of the deterministic system,and the rumor tends to extinction;when R_(0)>1,the rumor will keep spreading and the system will oscillate randomly near the rumor equilibrium point of the deterministic system.The results show that the oscillation amplitude is related to the disturbance of the system.In addition,increasing media coverage can effectively reduce the final spread of rumors.Finally,the above results are verified by numerical simulation.

Dynamics and near-optimal control in a stochastic rumor propagation model incorporating media coverage and Lévy noise

The appearance of rumors intensifies people’s panic and affects social stability. How to control the spread of rumors has become an important issue which is worth studying. In order to more accurately reflect the actual situation in the real world, a stochastic model incorporating media coverage and Lévy noise is proposed to describe the dynamic process of rumor propagation. By introducing two control strategies of popular science education and media coverage in an emergency event, an near-optimal control problem that minimizes the influence and control cost of rumor propagation is proposed.Sufficient conditions for near-optimal control of the model are established by using a Hamiltonian function. Then the necessary conditions for near-optimal control are obtained by using the Pontryagin maximum principle. Finally, the effect of popular science education, media coverage and Lévy noise on rumor propagation process control is verified by numerical simulation.

Ergodic stationary distribution of a stochastic rumor propagation model with general incidence function

In daily lives,when emergencies occur,rumors will spread widely on the internet.However,it is quite difficult for the netizens to distinguish the truth of the information.The main reasons are the uncertainty of netizens’behavior and attitude,which make the transmission rates of these information among social network groups be not fixed.In this paper,we propose a stochastic rumor propagation model with general incidence function.The model can be described by a stochastic differential equation.Applying the Khasminskii method via a suitable construction of Lyapunov function,we first prove the existence of a unique solution for the stochastic model with probability one.Then we show the existence of a unique ergodic stationary distribution of the rumor model,which exhibits the ergodicity.We also provide some numerical simulations to support our theoretical results.The numerical results give us some possible methods to control rumor propagation.Firstly,increasing noise intensity can effectively reduce rumor propagation when R_(0)>1That is,after rumors spread widely on social network platforms,government intervention and authoritative media coverage will interfere with netizens’opinions,thus reducing the degree of rumor propagation.Secondly,speed up the rumor refutation,intensify efforts to refute rumors,and improve the scientific quality of netizen(i.e.,increase the value ofβand decrease the value ofαandγ),which can effectively curb the rumor propagation.

Considering individual game behavior and time delay in the multi-channel rumor propagation model

The rapid development of the Internet has accelerated the spread of rumors,posing challenges to social cohesion and stability.To address this,a multi-channel rumor propagation model incorporating individual game behavior and time delay is proposed.It depicts individuals strategically choosing propagation channels in the rumor spread process,capturing real-world intricacies more faithfully.Specifically,the model allowing spreaders to choose between text and video information base channels.Strategy adoption hinges on benefits versus costs,with payoffs dictating strategy and the propagation process determining an individual's state.By theoretical analysis of the model,the propagation threshold and equilibrium points are obtained.Then the stability of the model is further demonstrated based on Routh-Hurwitz judgment and Descartes'Rule of Signs.Numerical simulations are conducted to verify the correctness of the theoretical results and the sensitivity of the model to key parameters.The outcomes reveal that increasing the propagation cost of spreaders can significantly curb the spread of rumors.In contrast to the classical ISR model,rumors spread faster and more widely in the improved multi-channel rumor propagation model in this paper,which is a feature more aligned with real-world scenarios.Finally,the validity and predictive ability of the model are verified by using real rumor propagation data sets,indicating that the improved multi-channel rumor propagation model has good practical application and predictive value.

Stability and optimal control for delayed rumor-spreading model with nonlinear incidence over heterogeneous networks

On the multilingual online social networks of global information sharing,the wanton spread of rumors has an enormous negative impact on people's lives.Thus,it is essential to explore the rumor-spreading rules in multilingual environment and formulate corresponding control strategies to reduce the harm caused by rumor propagation.In this paper,considering the multilingual environment and intervention mechanism in the rumor-spreading process,an improved ignorants–spreaders-1–spreaders-2–removers(I2SR)rumor-spreading model with time delay and the nonlinear incidence is established in heterogeneous networks.Firstly,based on the mean-field equations corresponding to the model,the basic reproduction number is derived to ensure the existence of rumor-spreading equilibrium.Secondly,by applying Lyapunov stability theory and graph theory,the global stability of rumor-spreading equilibrium is analyzed in detail.In particular,aiming at the lowest control cost,the optimal control scheme is designed to optimize the intervention mechanism,and the optimal control conditions are derived using the Pontryagin's minimum principle.Finally,some illustrative examples are provided to verify the effectiveness of the theoretical results.The results show that optimizing the intervention mechanism can effectively reduce the densities of spreaders-1 and spreaders-2 within the expected time,which provides guiding insights for public opinion managers to control rumors.

Dynamics of organizational rumor communication on connecting multi-small-world networks

We study the dynamics of an epidemic-like model for the spread of a rumor on a connecting multi-small-world- network （CM-SWN） model, which represents organizational communication in the real world. It has been shown that this model exhibits a transition between regimes of localization and propagation at a finite value of network randomness. Here, by numerical means, we perform a quantitative characterization of the evolution in the three groups under two evolution rules, namely the conformity and obeying principles. The variant of a dynamic CM-SWN, where the quenched disorder of small-world networks is replaced by randomly changing connections between individuals in a single network and stable connection by star nodes between networks, is also analysed in detail and compared with a mean-field approximation.

Study of Turing patterns in a SI reaction-diffusion propagation system based on network and non-network environments

The study of rumor propagation dynamics is of great significance to reduce.false news and ensure the authenticity of news information.In this paper,a SI reaction-diffusion rumor propagation model with nonlinear saturation incidence is studied.First,through stability analysis,we obtain the conditions for the existence and local stability of the positive equilibrium point.By selecting suitable variable as the control parameter,the critical value of Turing bifurcation and the existence theorem of Turing bifurcation are obtained.Then,using the above theorem and multi-scale standard analysis,the expression of amplitude equation around Turing bifurcation point is obtained.By analyzing the amplitude equation,different types of Turing pattern are divided such as uniform steady-state mode,hexagonal mode,stripe mode and mixed structure mode.Further,in the numerical simulation part,by observing different patterns corresponding to different values of control variable,the correctness of the theory is verified.Finally,the effects of different network structures on patterns are investigated.The results show that there are significant differences in the distribution of users on different network structures.

Lyapunov functions for studying global asymptotic stability of two rumor spreading models

In a previous work(2018,Commun.Theor.Phys.70,795–802),a new compartment model for the spreading of rumors was introduced and analyzed.However,only the local asymptotic stability of this model was discussed.In the present work,we first provide a rigorous mathematical analysis for the global asymptotic stability(GAS)of the above-mentioned rumor spreading model.By constructing suitable Lyapunov candidate functions,we obtain the GAS of a rumor-free(boundary)equilibrium point and a unique rumor-spreading(positive)equilibrium point.After that,we utilize the approach based on the Lyapunov candidate functions to study the GAS of another rumor spreading model with control strategies,which was proposed in(2022,Physica A 606,128157).As an important consequence,the GAS of the rumor spreading model with control strategies is determined fully without resorting to technical hypotheses used in the benchmark work.Lastly,the theoretical findings are supported by a set of illustrative numerical examples.The obtained results not only improve the ones constructed in the two abovementioned benchmark papers but also can be extended to study the global dynamics of other rumor propagation models in the context of both integer-order and fractional-order derivatives.

Modeling the dynamics of information propagation in the temporal and spatial environment

In this paper, we try to establish a non-smooth susceptible–infected–recovered(SIR) rumor propagation model based on time and space dimensions. First of all, we prove the existence and uniqueness of the solution. Secondly, we divide the system into two parts and discuss the existence of equilibrium points for each of them. For the left part, we define R_(0) to study the relationship between R_(0) and the existence of equilibrium points. For the right part, we classify many different cases by discussing the coefficients of the equilibrium point equation. Then, on this basis, we perform a bifurcation analysis of the non-spatial system and find conditions that lead to the existence of saddle-node bifurcation. Further, we consider the effect of diffusion. We specifically analyze the stability of equilibrium points. In addition, we analyze the Turing instability and Hopf bifurcation occurring at some equilibrium points. According to the Lyapunov number, we also determine the direction of the bifurcation. When I = I_(c), we discuss conditions for the existence of discontinuous Hopf bifurcation. Finally, through numerical simulations and combined with the practical meaning of the parameters, we prove the correctness of the previous theoretical theorem.

PREVENTING RUMOR SPREADING ON SMALL-WORLD NETWORKS

Since the spreading of harmful rumors can deeply endanger a society, it is valuable to investigate strategies that can efficiently prevent hazardous rumor propagation. To conduct this investigation, the authors modify the SIR model to describe rumor propagation on networks, and apply two major immunization strategies, namely, the random immunization and the targeted immunization to the rumor model on a small-world network. The authors find that when the average degree of the network is small, both two strategies are effective and when the average degree is large, neither strategy is efficient in preventing rumor propagation. In the latter case, the authors propose a new strategy by decreasing the credibility of the rumor and applying either the random or the targeted immunization at the same time. Numerical simulations indicate that this strategy is effective in preventing rumor spreading on the small-world network with large average degree.

Rumor Spreading Model with Immunization Strategy and Delay Time on Homogeneous Networks

In order to prevent and control the spread of rumors, the implementation of immunization strategies for ignorant individuals is very necessary, where the immunization usually means letting them learn the truth of rumors.Considering the facts that there is always a delay time between rumor spreading and implementing immunization, and that the truth of rumors can also be spread out, this paper constructs a novel susceptible-infected-removed(SIR) model.The propagation dynamical behaviors of the SIR model on homogeneous networks are investigated by using the meanfield theory and the Monte Carlo method. Research shows that the greater the delay time, the worse the immune effect of the immunization strategy. It is also found that the spread of the truth can inhibit to some extent the propagation of rumors, and the trend will become more obvious with the increase of reliability of the truth. Moreover, under the influence of delay time, the existence of nodes' identification force still slightly reduces the propagation degree of rumors.

Analysis on Impact of Rumors on Electricity Market Operations with Volatile Renewables

In recent years,rumors have been shown to have a significant impact on individual and societal activities.As renewables play an increasingly significant role in electricity markets,certain rumors may deviate the bidding behavior of market entities and eventually affect the performance of market operations.In this study,we attempt to reveal the general threats caused by rumors in the context of day-ahead electricity markets considering the integration of volatile renewables.First,we model the propagation of rumors in the societal system considering the weight of propagation resistance,which principally reflects the communication accessibility of market entities.Second,we develop an integrated two-layer network model to uncover the inherent coupling mechanism between market operations and rumor propagation.In particular,the role of electricity market operations on rumor propagation is characterized by changes in the truthfulness of rumors associated with electricity prices.The rumors,in turn,affect the bidding quantities of market entities in electricity market operations.Finally,numerical experiments are conducted on modified IEEE 6-bus and 118-bus systems.The results demonstrate the potential threats of rumors to electricity market operations with different penetration levels of renewables.