Propagation Dynamics Of Complex Networks With Low-order To High-order Structures

Recently,the propagation dynamics of complex networks has become a research hotspot due to propagation phenomena in the real world.By establishing mathematical models,the complex propagation process and various dynamical behaviors are investigated,thereby controlling dynamical processes.In complex networks,there are not only low-order structures of nodes and edges,but also widely high-order structures.Under various structures,different dynamical behaviors appear in the propagation of complex networks.Therefore,in this thesis,based on complex networks with low-order to highorder structures,two kinds of classical propagation dynamics including social contagion and epidemic spreading are investigated in different situations.Reasonable mathematical propagation models are proposed,and smooth evolution equations by the message passing method or the mean-field method are established,whose accuracy is also verified.The changes of dynamical behaviors caused by the various factors are analyzed,especially producing multi-type bistable in epidemic spreading due to the high-order structures.Furthermore,in order to study the propagation dynamics of disease under the practical strategy,a threshold policy is implemented in the epidemic spreading,and a non-smooth dynamical system is accordingly established.A sliding mode appears in epidemic spreading,and more importantly,the combined effects of non-smooth and higher-order structures lead to the tristable state.The results in this thesis enrich the models and theoretical results of social contagion and epidemic spreading.The effects of the network structure,heterogeneity and threshold policy on propagation dynamics are expanded,which provide a theoretical basis for formulating effective control strategies of disease.The main research texts of our thesis are as follows:First,in order to investigate the effects of the trust probability on propagation dynamics of social contagion,the trust probability with two influence factors including node degree and the number of common neighbors is proposed from the view of network topology in the thesis.And we concretely give the expression of the trust probability form a node to each neighbor.The message passing approach is adopted to formulate the state evolution equations of each node on the basis of network topology.The theoretical results agree with the numerical results well,which shows that the message passing equations can accurately describe the social contagion with the trust probability.The results of numerical simulations indicate that the trust probability can increase the epidemic threshold,and the final behavior adoption size decreases when the range of the trust probability increases.Notably,the number of common neighbors as an influence factor of the trust probability is able to increase the final behavior adoption size,while node degree takes the opposite effect.Then,a coupled propagation model on a double-layer network is set up to investigate the interaction of social contagion and epidemic spreading.The threshold model and classical susceptible-infected-recovered(SIR)model are adopted to describe social contagion and epidemic spreading,respectively.The coupling interaction is that if one adopts the behavior about disease,people will reduce the contact with neighbors to prevent the disease by setting a inhibiting factor.While once one is infected with disease,the individual can automatically get a piece of information about the behavior.In addition,the assumption in previous studies of coupled propagation dynamics that the time intervals of contact and recovery follow the exponential distribution is extended to arbitrary distribution.Further,the evolution equations in a finite double-layer network and a large scale random double-layer network are given via the message passing method.Numerical simulations are conducted to compare the theoretical results with the results of the Monte Carlo simulations,and the effects of the adoption threshold and the inhibiting factor on the social contagion and epidemic spreading are explored.On this basis,a threshold policy is implemented for epidemic spreading,and non-smooth message passing equations are established.The threshold policy makes the density of infected nodes keep a certain value at some time,thereby appearing the sliding mode in the evolution of epidemic spreading.Further,based on higher-order interactions in epidemic spreading and actual factors of the birth and death,a simplicial empty-susceptible-infected(ESI)epidemic model is proposed in this thesis.On this basis,the quenched mean-field probability equations for each site in the empty,susceptible and infected states are formulated,which form a highdimensional differential system.To theoretically analyze of the effects of the birth and death on the dynamics,the mean-field system is obtained by using the mean-field method to reduce the dimension,and its dynamics is analyzed.The results of theoretical analysis suggest that the birth and death rates influence the existence and stability of equilibria,as well as the appearance of a bistable state,which are then confirmed in numerical simulations on the empirical and synthetic networks.In addition,we find that another type of the bistable state in which a periodic outbreak state coexists with a stable disease-free state also emerges when the birth and death rates and other parameters satisfy the certain conditions.Finally,the results of numerical simulations illustrate that the birth and death rates shift the density of infected nodes in the stationary state and the outbreak threshold,which are also verified by sensitivity analysis for the proposed model.Finally,a threshold policy is introduced into simplical epidemic spreading with birth and death.A simplicial empty-susceptible-infected(ESI)peidemic model with a threshold policy is proposed.The probability evolution equations of each site in the explicit network are established by the quenched mean-field method,which is a non-smooth differential system due to the threshold policy.Remarkably,under the combined action of non-smooth and high-order structures,a tristable state is observed in empirical social networks,which is consistent with the coexistence of three stable equilibria by analysis of the mean-field system.The theoretical analysis of the mean-field system indicates that a sliding mode exists,which is confirmed by simulations of empirical social networks.The system is divided into the free and control subsystems by the threshold policy.Both subsystems admit a stable disease-free equilibrium and a stable endemic equilibrium,as well as coexistence of a stable disease-free equilibrium and a stable pseudo equilibrium in the system,thereby occurring three types of the bistable state under the threshold policy with different critical levels.