Deriving Exact Mathematical Models of Malware Based on Random Propagation

Rodrigo Matos Carnier (), Yue Li, Yasutaka Fujimoto and Junji Shikata
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Rodrigo Matos Carnier: Information Systems Architecture Research Division, National Institute of Informatics, 2-1-2 Hitotsubashi, Chiyoda City, Tokyo 101-8430, Japan
Yue Li: Department of Electrical and Computer Engineering, Yokohama National University, 79-5 Tokiwadai, Hodogaya Ward, Yokohama 240-8501, Japan
Yasutaka Fujimoto: Department of Electrical and Computer Engineering, Yokohama National University, 79-5 Tokiwadai, Hodogaya Ward, Yokohama 240-8501, Japan
Junji Shikata: Graduate School of Environment and Information Sciences, Yokohama National University, 79-5 Tokiwadai, Hodogaya Ward, Yokohama 240-8501, Japan

Mathematics, 2024, vol. 12, issue 6, 1-28

Abstract: The advent of the Internet of Things brought a new age of interconnected device functionality, ranging from personal devices and smart houses to industrial control systems. However, increased security risks have emerged in its wake, in particular self-replicating malware that exploits weak device security. Studies modeling malware epidemics aim to predict malware behavior in essential ways, usually assuming a number of simplifications, but they invariably simplify the single most important subdynamics of malware: random propagation. In our previous work, we derived and presented the first exact mathematical model of random propagation, defined as the subdynamics of propagation of a malware model. The propagation dynamics were derived for the SIS model in discrete form. In this work, we generalize the methodology of derivation and extend it to any Markov chain model of malware based on random propagation. We also propose a second method of derivation based on modifying the simplest form of the model and adjusting it for more complex models. We validated the two methodologies on three malware models, using simulations to confirm the exactness of the propagation dynamics. Stochastic errors of less than 0.2% were found in all simulations. In comparison, the standard nonlinear model of propagation (present in ∼95% of studies) has an average error of 5% and a maximum of 9.88% against simulations. Moreover, our model has a low mathematical trade-off of only two additional operations, being a proper substitute to the standard literature model whenever the dynamical equations are solved numerically.

Keywords: malware model; random propagation; Markov chain (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2024
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