Preventive and Reactive Cyber Defense Dynamics Is Globally Stable

The recently proposed 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 attacks and the various kinds of cyber defenses that take place in complex networks. In this paper, we study a particular kind of cybersecurity dynamics caused by the interactions between two classes of attacks (called push-based attacks and pull-based attacks) and two classes of defenses (called preventive and reactive defenses). The dynamics was previously shown to be globally stable in a special regime of the parameter universe of a model with node-independent and edge-independent parameters, but little is known beyond this regime. In this paper, we prove that the dynamics is globally stable in the entire parameter universe of a more general model with node-dependent and edge-dependent parameters. This means that the dynamics always converges to a unique equilibrium. We also prove that the dynamics converges exponentially to the equilibrium except for a particular parameter regime, in which the dynamics converges polynomially. Since it is often difficult to compute the equilibrium, we propose bounds of the equilibrium and numerically show that these bounds are tighter than those proposed in the literature.