The Influence Of Degree Heterogeneity On Synergistic Epidemic Spreading

From nature to human society,many spreading phenomena can be described as the epidemic-like spreading phenomena in networked systems.For example,the propagation of rumors and information through online social networks,the spread of diseases in human contact networks,the diffusion of malware and virus on computer networks,the spread of cascade failure in power grid networks and the neural activities in the human brain.The research on the spreading dynamics in complex networks can provide useful insights on the spreading processes,thus laying crucial foundation for the control of the spread of diseases,products and information.Synergistic effect,known as the mechanism which describes that the spreading can be enhanced or hampered by nodes’ cooperation,is ubiquitous in the realistic spreading processes and is also one of the essential questions of the research of the system theory.For example,the explosive diffusion of short videos,buzzwords and products on online networks might be promoted by the synergistic effect.The properties of complex network structure could have profound influences on the spreading dynamics,and in particular the influence of degree heterogeneity is a hot topic in the research community.Previous studies have shown that the synergistic spreading on homogeneous networks usually displays an explosive transition while the one on heterogeneous networks follows a continuous epidemic transition.However,while the network structure is continuously varying from the homogeneous to the heterogeneous structures or from the heterogeneous to the homogeneous structures,how the epidemic transition changes toward continuous or explosive ones remains ambiguous.Based on a mixed network model,this paper achieves continuous regulation of network structure heterogeneity,and studies this change of epidemic transition through modeling,theoretical analysis and simulation,which is of great significance for understanding the impact of network structure and synergistic effects on propagation phenomena.In the first part of this paper,we construct a synergistic spreading model based on the findings of previous studies and generate networks with various levels of degree heterogeneity using the mixed network model.We simulate the synergistic spreading model on these networks and numerically analyze the influence of degree heterogeneity and structural heterogeneity on the synergistic spreading processes.In the realist spreading processes,there exist many interacting and undistinguishable higher-order contagions,and they will produce a mean-field synergistic effect globally.Therefore,we introduce a global synergistic effect into the classic Susceptible-Infected-Susceptible(SIS)model and construct the synergistic SIS model which can better characterize the realistic spreading processes.Moreover,to continuously adjust the degree heterogeneity,we use the mixed network model to generate networks with various levels of degree heterogeneities.Detailed simulations show that the critical threshold of synergistic strength for the explosive transition increases with degree heterogeneity,i.e.,strong degree heterogeneity will hinder the appearance of explosive transition.Moreover,the synergistic spreading processes on heterogeneous networks will exhibit explosive transition as well when the synergy strength is sufficiently strong.In the second part of this paper,we employ the heterogeneous meanfield theory to describe the synergistic spreading on networks,aiming to gain the intuitive insights on the influence of degree heterogeneity analytically.We first apply the first order approximation to simplify the heterogeneous mean-field equation.Then we obtain the epidemic threshold(s)and critical synergistic strength by solving the simplified selfconsistent equations,with which we analyze the influences of degree heterogeneity on the synergistic spreading.Analytical results show that the critical synergistic strength would increase as the value of degree heterogeneity increase,indicating that strong degree heterogeneity could hinder the appearance of explosive spreading transition.Based on the analytic results of first order approximation,we further obtain a more accurate critical synergy strength by solving the self-consistent equations.The results of simulations are consistent with the analytical ones.Thus,our analytical results can help to understand the influence of degree heterogeneity on synergistic spreading.In conclusion,we analyze the influence of degree heterogeneity on synergistic spreading systematically.We show that strong degree heterogeneity will hinder the appearance of explosive epidemic transition and how the epidemic transition changes toward an explosive or continuous one with decreasing or increasing degree heterogeneity.We also provide a theoretical framework for synergistic spreading in this paper,which can provide useful insights on synergistic spreading processes on complex networks.