[Paper Review] Preventive and Reactive Cyber Defense Dynamics with Ergodic Time-dependent Parameters Is Globally Attractive

Yujuan Han, Wenlian Lu+1 more|arXiv (Cornell University)|Jan 22, 2020

Opinion Dynamics and Social Influence44 references4 citations

TL;DR

This paper establishes global attractivity for preventive and reactive cyber defense dynamics with ergodic time-dependent parameters using skew-product semi-flows and the multiplicative ergodic theorem. It proves that such dynamics converge globally when parameters are ergodic, extending prior results on time-independent and periodic cases, and suggests subhomogeneity and ergodicity may be necessary for attractivity.

ABSTRACT

Cybersecurity dynamics is a mathematical approach to modeling and analyzing cyber attack-defense interactions in networks. In this paper, we advance the state-of-the-art in characterizing one kind of cybersecurity dynamics, known as preventive and reactive cyber defense dynamics, which is a family of highly nonlinear system models. We prove that this dynamics in its general form with time-dependent parameters is globally attractive when the time-dependent parameters are ergodic, and is (almost) periodic when the time-dependent parameters have the stronger properties of being (almost) periodic. Our results supersede the state-of-the-art ones, including that the same type of dynamics but with time-independent parameters is globally convergent.

Motivation & Objective

• To extend the global convergence results of preventive and reactive cyber defense dynamics from time-independent to time-dependent parameters.
• To address the limitations of prior models that assume constant or periodic parameters, which restrict real-world applicability.
• To investigate whether ergodicity and subhomogeneity are necessary conditions for global attractivity in non-autonomous cyber defense dynamics.
• To provide a rigorous theoretical foundation for analyzing complex, time-varying cyber attack-defense interactions using dynamical systems theory.
• To establish a framework that supersedes existing results on time-independent, periodic, and special cases of the unified cyber defense model.

Proposed method

• Formulates the unified preventive and reactive cyber defense dynamics as a non-autonomous dynamical system with time-dependent parameters.
• Applies the skew-product semi-flow approach to analyze the long-term behavior of the system under time-varying parameters.
• Employs the multiplicative ergodic theorem to study the asymptotic behavior of the system when parameters are ergodic.
• Uses subhomogeneity and ergodicity as key structural assumptions to prove global attractivity.
• Conducts numerical experiments on Erdős–Rényi random graphs to test the necessity of subhomogeneity and ergodicity.
• Compares the behavior of the ∑-model under non-subhomogeneous and non-ergodic parameter functions to demonstrate failure of global attractivity.

Experimental results

Research questions

• RQ1 Under what conditions is preventive and reactive cyber defense dynamics globally attractive when parameters are time-dependent?
• RQ2 Is ergodicity of time-dependent parameters a necessary condition for global attractivity in this class of cyber defense dynamics?
• RQ3 Can the global attractivity result be extended to non-autonomous systems where parameters vary in a non-periodic but ergodic manner?
• RQ4 How do subhomogeneity and ergodicity influence the long-term convergence behavior of the system?
• RQ5 What is the boundary between analytically tractable and intractable cyber defense dynamics under time-varying parameters?

Key findings

• The unified preventive and reactive cyber defense dynamics with ergodic time-dependent parameters is globally attractive, extending prior results on time-independent and periodic cases.
• When time-dependent parameters are (almost) periodic, the dynamics is (almost) periodic, indicating structured long-term behavior.
• Numerical experiments show that violating subhomogeneity in the defense function leads to non-global attractivity, suggesting subhomogeneity may be necessary.
• Numerical results also indicate that non-ergodic parameters can break global attractivity, hinting that ergodicity may be a necessary condition.
• The global attractor is time-dependent, making standard eigenvalue analysis of the Jacobian inapplicable, thus requiring advanced tools like the multiplicative ergodic theorem.
• The theoretical framework supersedes previous results on special cases, including time-independent, periodic, and specific ∏- and ∑-models.