Impact Of Inter-layer Traveling On Spreading Dynamics In Multilayer Networks

The introduction of small-world network and scale-free network models suggests that the research of graph theory has entered the era of complex network.After more than 20 years of development,the research of complex network has attracted the attention of scholars in the fields of mathematics,physics,computer,information,sociology and biology,and becomes an important inter-disciplinary subject.It begins to focus on static and isolated networks.Nevertheless,there are complex and diverse individual interaction in most natural and engineering systems,and such systems should be modeled by multilayer network coupled from multiple layers.For example,users spread information in a variety of social platforms,such as Wechat,Weibo,and Tiktok.Their interactions among these multiple social platforms greatly accelerate the speed of information dissemination.There are many ways of transportation for both human beings and goods,such as highway,railway,shipping and aviation.Using reasonably transportation networks can effectively improve transportation efficiency.Therefore,it is one of hot topics in the field of complex networks to establish the multilayer network theory for revealing the broader structural characteristics and exploring the dynamics processes on them.Epidemic spreading is one of the top problems perplexing mankind.Each pandemic will bring enormous health threats and economic losses.Our work focuses on the study of epidemic spreading dynamics on multilayer networks.We propose a two-layer network model where a fraction of nodes are inter-layer travelers that can hop between layers,and study the impact of travelers’ inter-layer hopping on the epidemic spreading dynamical processes.The main contents are as follows:(1)We study the impact of the presence of travelers on spreading dynamics.We present a two-layer network model with travelers where the inter-layer coupling is carried by travelers’ inter-layer hopping between the matching nodes in two layers at a certain rate,which is different from the fixed couplings in previous systems.Firstly,we study the impact of inter-layer hopping on susceptible-infected-removed(SIR)type epidemic dynamics on a double-layer symmetric networked system.We find that due to the “hole” effect brought by travelers,the actual average degree of each layer is reduced,which will suppress the epidemic spreading on the multilayer network.Then we focus on the asymmetric two-layer networks.The hopping behaviors of travelers bring the inter-layer coupling,which promotes the epidemic spreading in the sparse layer.Due to the opposite effects on the spreading dynamics introduced by inter-layer coupling and “hole” effect,the outbreak size and threshold of epidemic on sparse layer show interesting non-monotonic changes with the increase of the proportion of travelers.This means that travelers have a “double-edged sword” effect on the epidemic spreading on sparse layer.Extensive numerical simulations on empirical networks verified the universality of the above results.We develop a theoretical framework based on the quenched mean-field approximation to describe the spreading dynamic on the multilayer networks,which agree well with the simulation results on both synthetic networks and real networks.Finally,we observe the “multiple bursts” in sparse layer on the model,which helps us to understand the multiple recurrence phenomenon in real epidemic spreading.(2)We study the impact of hopping characteristics of inter-layer travelers on epidemic spreading in multilayer networks.The activity and preference of travelers’ inter-layer hopping are defined in the above double-layer network model,and their impacts on spreading dynamics is studied.Extensive numerical simulations and theoretical analyses indicate that the activity of travelers determines the coupling strength between layers in the spreading processes.As the travelers’ hopping activity increases,the spreading on the dense and sparse layers is restrained and promoted,respectively.The travelers’ hopping preference has a more subtle impact on the spread of epidemic on each layer.Hopping preference determines the distribution of travelers and the active topology in the layers,and then affects the dynamic process in each layer and the coupling between layers.As the preference of hopping into sparse layer increases,the epidemic spreading process of dense layer is restrained.The spreading in the sparse layers is first suppressed and then promoted,where the infection density and threshold of the epidemic have a minimal value and maximal value,respectively.This study provides theoretical supports for the prevention and control of epidemic in the multilayer networks.(3)The null model method is used to study the influence of the inter-layer structural correlation on the structural characteristics and dynamic processes of multilayer networks.We establish a message-telephone double-layer network based on the real communication data,and assume that the users using two communication modes at the same time are the travelers of the double-layer network.Firstly,the average active degrees of the two layers decreases with the increase of the inter-layer correlation coefficient,which inhibits the spreading dynamics in both layers.However,the increase of inter-layer correlation promotes the disease diffusion from the dense layer to the sparse layer,which significantly increases the outbreak size in the sparse layer.Under the dual effects of the decrease of the average activity within the layer and the strengthened virus diffusion between two layers,the outbreak size in the sparse layer shows a non-monotonic change of first decreasing and then increasing with the increase of activity’s inter-layer correlation coefficient.This study suggests that in the spreading process,if the inter-layer correlation of travelers is assortative,the inter-layer travel restrictions can effectively suppress inter-layer diffusion and intra-layer spreading.In summary,we have proposed a two-layer network model which is coupled by the inter-layer hopping of inter-layer travelers,and established a theoretical framework to describe the impacts of travelers on the structure and dynamics of the two-layer network.Our research provides a new perspective and analytical method to understand the spreading dynamics on multilayer networks.It also provides a reference for the prediction and prevention of epidemics.