Non-Markovian Social Contagions On Complex Networks

Human activities are heavily dependent on the social networks such as infrastructure networks,online and offline communication networks,and the Internet.There are numerous dynamics on these networks,such as information diffusion,which causes extensive attention among scientists.For instance,empirical evidences show that social reinforcement is widespread in social contagions and thus a variety of complex contagion models are proposed,including both the susceptible-infected-recovered(SIR)type and susceptible-infected-susceptible(SIS)type of complex contagion models.The former can be used to describe the short-term rumor spreading processes,and the latter can be used to describe the spreading processes on the networks such as the stock markets and social communication networks,where a susceptible node can be infected and then recover to susceptible again.An assumption in many previous studies of contagion is that the contagion process is Markovian.However,most dynamic processes in the real world are often highly non-Markovian,such as company reorganization,web browsing,and new product promotion.In a non-Markovian process,the current state of a node depends not only on its most recent state but also on its previous states,which is memorable and thus makes the analytic treatment being of challenge.In the past decade,some progress has been made in the study of non-Markovian processes on spreading dynamics,but many key issues have not been resolved.In this dissertation,we focus our main attention on SIS type of complex contagion models to investigate the non-Markovian social contagion on complex networks.The main results are as follows:(1)Develop the theoretical framework of the spontaneous recovery system.First,we introduce the Markovian recovery(MR)model and the non-Markovian recovery(NMR)model: The spontaneous recovery model is a type of SIS threshold model,which is a complex contagion model.Either the Markovian model or the nonMarkovian model contains two types of failure-and-recovery scenarios: internal and external.In the MR model,failures due to internal and external causes will recover with different constant rates.In the NMR model,such a constant rate cannot be defined.We thus resort to the recovery time.In these two models,the former is an ideal scenario,while the latter is the realistic one.Most previous studies treated Markovian processes through either a mean-field theory or an effective degree approach.For non-Markovian processes,the mean-field approximation can still be applied,but it is necessary to invoke a higher-order theory such as the pairwise approximation analysis.In this chapter,we develop a set of pairwise approximation theory for the two models.To capture the memory effect of the NMR process,we express the model in terms of difference equations in the theory by decomposing the NMR process into a series of MR processes.Besides,the mean-field analysis shows that there is an equivalent relationship between the two models,which makes a solid theoretical foundation for understanding the influence of the non-Markovian recovery process on the failure propagation in the next chapter.(2)Reveal the significant influence of non-Markovian recovery on the failure propagation.Our study demonstrates that,in both the NMR and the MR models,the network can evolve into a low-failure or a high-failure state,with the latter corresponding to the undesired state of large scale failure.Both the mean-field and pairwise approximation theories are capable of predicting the dynamical behaviors of failure propagation,and the performances of the theories are gauged by simulation results,revealing that the more laborious pairwise approximation gives results in better quantitative agreement with the numerics.Next,we study the behavior of MR model.We find that the system exhibits phenomena such as first-order phase transition and hysteresis.Through meanfield analysis,we derive the analytical forms of low-failure and high-failure branches.Furthermore,our systematic computations on different complex networks and two types of theoretical analyses uncover a striking phenomenon: when the initial infected nodes take the same time to recover,the non-Markovian memory effect in the nodal recovery can counter-intuitively make the network more resilient against large scale failures.This is the most important phenomenon found in this chapter.Subsequently,we analyse the results.Finally,We carry out a systematic study of the effects of Markovian versus non-Markovian recovery on network synchronization by using the paradigmatic Kuramoto network model,with the main finding that nonMarkovian recovery makes the network more resilient against large-scale breakdown of synchronization.In natural systems,the intrinsic non-Markovian characteristic of nodal recovery may thus be one reason for their resilience.In engineering design,incorporating certain non-Markovian features into the network may be beneficial to equipping it with a strong resilient capability to resist catastrophic failures.(3)Develop the model of the higher-order non-Markovian social contagion on the simplicial complexes.In the above two chapters,we have studied the spontaneous recovery model,which is a threshold model incorporating the social reinforcement.However,the shortcoming of the threshold model is that the interaction between nodes is transmitted through the edges(low-dimensional simplex structure),which cannot describe the interdependence of nodes in higher-order structures(e.g.,high-dimensional simplex structures such as triangles and quadrilaterals)in the real world.Therefore,a higherorder Markovian social contagion model on simplicial complexes was proposed.We do further work on this aspect.First,we extend the model to the case of higher-order non-Markovian social contagion and consider the general situation where the infection and recovery are both non-Markovian processes.Then,we develop the non-Markovian quenched mean-field theory and perform numerical simulations,which reveal that the theory is capable to predict the results of steady state.After that,our theoretical analysis indicates that there is an equivalent relationship between the higher-order nonMarkovian social contagion and the higher-order Markovian social contagion,which is verified by the simulation results.Finally,we study the influence of higher-order non-Markovian recovery processes on the propagation.We consider the general situation where the processes of infection and recovery are both non-Markovian,and find that when the initial infected nodes take closer time to recover,the non-Markovian recovery can make the network more resilient against very large-scale failures.Furthermore,when the initial infected nodes take shorter and closer time to recover,the non-Markovian recovery can make the network more resilient against relatively smallscale failures.Subsequently,we analyse the results.The results help us better understand the behavior of higher-order non-Markovian social contagion in the real world.In conclusion,this dissertation proposes and develops the non-Markovian social contagion models and theoretical frameworks,respectively.The dissertation investigates the non-Markovian social contagions and provides the new ideas and methods for the future study of non-Markovian social contagion on complex networks.