Preventive and Reactive Cyber Defense Dynamics Is Globally Stable

TL;DR

This paper proves that a model of cybersecurity dynamics involving preventive and reactive defenses against cyber attacks is globally stable across all parameters, ensuring predictable long-term behavior.

Contribution

It extends previous stability results to the entire parameter space and introduces tighter bounds for the equilibrium, enhancing understanding of cybersecurity dynamics.

Findings

The dynamics always converge to a unique equilibrium.

Convergence is exponential except in a special case where it is polynomial.

New bounds for the equilibrium are tighter than existing ones.

Abstract

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.