Stochastic and Percolating Models of Blocking Computer Networks Dynamics during Distribution of Epidemics of Evolutionary Computer Viruses

Abstract

The paper presents a complex model of the dynamics of virus epidemies propagation in computer networks, based on topological properties of computer networks and mechanisms of the viruses spread. On one hand, this model is based on the use of percolation theory methods, which makes it possible to determine such structural-information characteristics of networks as the dependence of the percolation threshold on the average number of connections per one node (network density). On the other hand, the dynamic processes of stochastic propagation in computer networks of evolving viruses are observed when anti-virus programs become outdated and postponed. The paper discusses the concept of percolation threshold, provides an equation for the dependence of the percolation threshold of a network on its density obtained by analyzing numerical simulation data. The dynamics of virus epidemies were studied through two approaches. The first one is based on the description of transition diagrams between states of nodes, after which a system of kinetic differential equations for the virus epidemies is constructed. The second is based on considering the probabilities of transitions between possible states of the entire network. A second-order differential equation is obtained in this article, and a boundary value problem is formulated. Its solution describes the dependence of the network blocking probability on the blocking probability of an individual node. The solution also makes it possible to estimate the time required to reach the percolation threshold. The model incorporates the evolutionary properties of viruses: previously immunized or disinfected nodes can be infected again after a certain time interval. Besides, the model incorporates a lag of the anti-virus protection. Analysis of the solutions obtained for the models created shows the possibility of various modes of virus propagation. Moreover, with some sets of values of differential equation coefficients, an oscillating and almost periodic nature of virus epidemies is observed, which largely coincides with real observations.