Preventive and Reactive Cyber Defense Dynamics Is Globally Stable

The recently proposed {\em cybersecurity dynamics} approach aims to understand cybersecurity from a holistic perspective by modeling the evolution of the global cybersecurity state. These models describe the interactions between the various kinds of cyber defenses and the various kinds of cyber attacks. We study a particular kind of cybersecurity dynamics caused by the interactions between preventive and reactive defenses (e.g., filtering and malware detection) against push- and pull-based cyber attacks (e.g., malware spreading and "drive-by download" attacks). The dynamics was previously shown to be globally stable in a {\em special} regime of the parameter universe, but little is known beyond this special regime. In this paper, we resolve an open problem in this domain by proving that the dynamics is globally stable in the {\em entire} parameter universe (i.e., the dynamics always converges to a unique equilibrium). We discuss the cybersecurity meanings and implications of this theoretic result. We also prove that the dynamics converges {\em exponentially} to the equilibrium except for a special parameter regime, in which case the dynamics converges {\em polynomially}. Since it is often difficult to compute the equilibrium, we propose new bounds of the equilibrium and numerically show that these bounds are tighter than those proposed in the literature.