  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.
