Topological Structures And Analysis Of Transmission Dynamics On Networks

The security of military information networks determines the direction of the future war,and the spreading of virus on networks is one of the important factors influencing the securityof network information. To make effective prevention and control strategies, it is needed tofind the inner mechanisms, research tools and theoretical methods of the network virusspreading. The rise of research on complex networks theory allows people to make scientificexplanations on the network phenomenons and their complexities, and provides a newtechnique for the information security of military networks. To do this, one can establish therelevant analysis models or prediction models on the real observed phenomenons in the social,military and other networks, and use the topological structures of networks to explain theirinner mechanisms. The research of complex networks mainly consists of the topologicalstructures of networks, the establishment of network mechanism models and the study ofdynamic behaviors on networks. This dissertation proposes a growth mechanism model ofnetwork and several network-based virus models, and investigates the methods ofcharacterization of topological structures of networks, the way to build and reconstruct thereal network mechanisms and the effect of topological structures of networks on the pattern ofvirus spreading. And the dynamical method, which used to study the spreading behaviors ofnetwork virus, provides a theoretical basis for the virus prevention strategies in the social andmilitary networks.In chapter1, an introduction to complex networks theory, the significance for researchand the relationships with modeling of virus spreading are given. The main contents includethe basic network concepts, the description of topological structures and three classicalmechanism models of networks. On this basis, the thinking and development history ofmodeling of virus spreading on complex networks and the difference between network-basedvirus models with the well-mixing virus models are also introduced. In chapter2, by introducing several kinds of representative network-based virus modelsand their main conclusions, we briefly sketch and review the development of the approach ofnetwork-based virus modeling, including PSV, moment closure, percolation theory andedge-based virus models.In chapter3, we propose and study a generic growth mechanism model of multiplex socialnetwork, which consists of two different layers representing friendship and contactrelationships respectively, and a newly arrived node to establish connections to existing nodesin each of the layers of the multiplex social network is a function of the degree of other nodesat all layers. The degree distributions on each of the layers are explored utilizing a mean-fieldmodel. Furthermore, we analyze and compare the degree distributions, the assortative ordisassortative properties and the clustering coefficients between the two layers of the networkby random simulations.In chapter4, to capture the phenomenon that symmetrical and asymmetrical contacts canoccur in the same contact network, we establish an SIS model for virus spreading onsemi-directed networks, the basic reproduction number is obtained by the local stability of thevirus-free equilibrium. And we proved the global stability of virus-free equilibrium andpositive equilibrium respectively. Furthermore, the effect of topological structures resultsfrom undirected and directed contacts on virus spreading is examined and compared.In chapter5, we give the global analysis of an SIS virus model on adaptive networks, inwhich how rewiring mechanism influences the virus dynamics is studied analytically. And weconclude that the topological structures of networks, caused by rewiring mechanism onadaptive networks, can lead to the occurrence of rich dynamics on virus spreading models,such as backward bifurcation, bistability and Hopf bifurcation.In chapter6, two types of network-based virus models are established, we investigate theeffect of multi-routes and the distribution of infectious period on the spreading of virus,respectively. First，a new virus spreading model with multiple routes was established, inwhich the transmission mechanisms of homogenous and heterogeneous can not occur on eachnode simultaneously. The uniqueness of the virus-free equilibrium is obtained by the limit system and Gershgorin disk theorem. And then according to the local stability of thevirus-free equilibrium，we obtain the basic reproduction number. In addition, different fromthe exponential distribution selected in the past, we propose a virus model with arbitrarydistributed infectious period on complex networks, and the global dynamic behaviors arestudied.