Modeling and analyzing the dynamic spreading of epidemic malware by a network eigenvalue method

This paper mainly focuses on studying the influence of network characteristics on malware spreading. Firstly, a generalized model with weakly-protected and strongly-protected susceptible nodes is developed by considering the possibility of an intruded node converting back to a weakly-protected susceptible one. The dynamics of the generalized compartmental model is intensively discussed and analyzed, deriving several sufficient conditions for its global stability. Following this work, a novel node-based model is newly proposed to describe malware propagation over an arbitrary connected network including synthesized and real networks. From a microscopic perspective, we establish the novel model by introducing several different variables for each node which describe the probabilities of a node locating at respective states. Our theoretical analysis shows that the largest eigenvalue of the propagating network is a key factor determining malware prevalence. Specifically, the range of the leading eigenvalue can be split into three subintervals in which malware approaches extinction very quickly, or tends to extinction, or persists, depending on into which subinterval the largest eigenvalue of the propagating network falls. Theoretically, the trivial equilibrium of our new node-based model is clearly proved to be exponentially globally stable when the maximum eigenvalue is less than a threshold. We also illustrate the predictive effectiveness of our model by designing some numerical simulations on some regular and scale-free networks. Consequently, we conclude that malware prevalence can be effectively prevented by properly adjusting the spreading network, e.g., reducing the number of nodes and deleting some edges, so that its maximum eigenvalue falls into the appropriate subinterval.