On discrete-time models of network worm propagation generated by quadratic operators
Abstract
In this paper we consider the discrete-time dynamical systems generated by network worm propagation models based on the theory of quadratic stochastic operators(QSO). This approach simultaneously solves two important problems: exploring of the QSO trajectory‘s behavior, we described the set of limit points, thereby completely solved the main problem of dynamical systems (i), we showed a new application of the theory QSOs in worm propagation modelling (ii). We demonstrated that proposed discrete-time biologically-inspired model represents also realistic picture of the worm propagation process and such analytical models can be used in decision of some problems of computer networks.
Keyword : network worms, propagation dynamics, modeling, quadratic stochastic operator, regular operator
How to Cite
Adilova, F., Jamilov, U., & Reinfelds, A. (2023). On discrete-time models of network worm propagation generated by quadratic operators. Mathematical Modelling and Analysis, 28(2), 194–217. https://doi.org/10.3846/mma.2023.15999
This work is licensed under a Creative Commons Attribution 4.0 International License.
References
E. Akin and V. Losert. Evolutionary dynamics of zero-sum games. Journal of Mathematical Biology, 20(3):231–258, 1984. https://doi.org/10.1007/BF00275987
L.J.S. Allen. Some discrete-time SI, SIR, and SIS epidemic models. Mathematical BioSciences, 124(1):83–105, 1994. https://doi.org/10.1016/0025-5564(94)90025-6
V.M. Barashkov and N.A. Zadorina. Analysis of two-stage mathematical models of computer distribution viruses. In A.B. Yegorov, V.Ya. Hentov and at al.(Eds.), Proc. of the 5th Intern. Conference Actual Problems Technical Sciences in Russia and abroad, Novosibirsk, Russia, 2018, pp. 96–102, 2018. (in Russian)
J. Blath, U.U. Jamilov(Zhamilov) and M. Scheutzow. (G,µ)- quadratic stochastic operators. Journal of Difference Equations and Applications, 20(8):1258–1267, 2014. https://doi.org/10.1080/10236198.2014.906591
D. Chumachenko, K. Chumachenko and S. Yakovlev. Intelligent simulation of network worm propagation using the code red as an example. Telecommunications and Radio Engineering, 78(5):443–464, 2019. https://doi.org/10.1615/TelecomRadEng.v78.i5.60
D. Chumachenko and S. Yakovlev. Development of deterministic models of malicious software distribution in heterogeneous networks. In 2019 3rd International Conference on Advanced Information and Communications Technologies (AICT), pp. 439–442, 2019. https://doi.org/10.1109/AIACT.2019.8847903
Á.M. del Rey. A SIR e-epidemic model for computer worms based on cellular automata. In C. Bielza, A. Salmerón and et al. (Eds.), Advances in Artificial Intelligence, pp. 228–238, Berlin, Heidelberg, 2013. Proceedings of 15th Conference of the Spanish Association for Artificial Intelligence, CAEPIA 2013 Madrid, Spain, September 17-20, 2013.
R.L. Devaney. An introduction to chaotic dynamical systems. Studies in Nonlinearity, Westview Press, Boulder, CO, 2003.
N.N. Ganikhodjaev, R.N. Ganikhodjaev and U.U. Jamilov. Quadratic stochastic operators and zero-sum game dynamics. Ergodic Theory and Dynamical Systems, 35(5):1443–1473, 2015. https://doi.org/10.1017/etds.2013.109
N.N. Ganikhodjaev, U.U. Jamilov and R.T. Mukhitdinov. On non - ergodic transformations on S3. Journal of Physics: Conference Series, 435(1):012005, 2013. https://doi.org/10.1088/1742-6596/435/1/012005
R.N. Ganikhodzhaev. Quadratic stochastic operators, Lyapunov functions and tournaments. Russian Academy of Sciences. Sbornik Mathematics, 76(2):489– 506, 1993. https://doi.org/10.1070/SM1993v076n02ABEH003423
R.N. Ganikhodzhaev. Map of fixed points and Lyapunov functions for one class of discrete dynamical systems. Math. Notes, 56(5):1125–1131, 1994. https://doi.org/10.1007/BF02274660
R.N. Ganikhodzhaev, F.M. Mukhamedov and U.A. Rozikov. Quadratic stochastic operators and processes: results and open problems. Infinite Dimensional Analysis, Quantum Probability and Related Topics, 14(2):279–335, 2011. https://doi.org/10.1142/S0219025711004365
U. Jamilov. On a family of strictly non-Volterra quadratic stochastic operators. Journal of Physics: Conference Series, 697(1):012013, 2016. https://doi.org/10.1088/1742-6596/697/1/012013
U.U. Jamilov. Quadratic stochastic operators corresponding to graphs. Lobachevskii Journal of Mathematics, 34(2):148–151, 2013. https://doi.org/10.1134/S1995080213020042
J. O. Kephart and S. R. White. Directed-graph epidemiological models of computer viruses. In Proc. of the 1991 IEEE Computer Society Symposium on Research in Security and Privacy, Oakland, California, pp. 343–359, 1991.
H. Kesten. Quadratic transformations: A model for population growth. I. Advances in Applied Probability, 2(1):1–82, 1970. https://doi.org/10.2307/3518344
I.V. Kotenko and V.V. Vorontsov. Analytical models of network worm propagation. In SPIIRAS Proceedings, volume 4, pp. 208–224, SPb., 2007. Nauka. https://doi.org/10.15622/sp.4.15 (in Russian)
Y.I. Lyubich. Mathematical structures in population genetics. vol. 22 of Biomathematics, Springer-Verlag, Berlin, 1992.
P. Van Mieghem. The n-intertwined SIS epidemic network model. Computing, 93(2):147–169, 2011. https://doi.org/10.1007/s00607-011-0155-y
F. Mukhamedov and A.F. Embong. On stable of b-bistochastic quadratic stochastic operators and associated nonhomogenous Markov chains. Linear and Multilinear Algebra, 66(1):1–21, 2018. https://doi.org/10.1080/03081087.2017.1281215
F. Mukhamedov and N. Ganikhodjaev. Quantum Quadratic Operators and Processes. vol. 2133, Lecture Notes in Mathematics, Springer-Verlag, Berlin, 2015. https://doi.org/10.1007/978-3-319-22837-2
F. Mukhamedov, M. Saburov and I. Qaralleh. On ξ(s)-quadratic stochastic operators on two-dimensional simplex and their behavior. Abst. Appl. Anal., 2013:942038, 2013. https://doi.org/10.1155/2013/942038
U.A. Rozikov and S. K. Shoyimardonov. Ocean ecosystem discrete time dynamics generated by l− Volterra operators. International Journal of Biomathematics, 12(2):1950015, 2019. https://doi.org/10.1142/S1793524519500153
U.A. Rozikov and S.S. Xudayarov. Quadratic non-stochastic operators: examples of splitted chaos. Annals of Functional Analysis, 13(1):17, 2022. https://doi.org/10.1007/s43034-021-00162-5
U.A. Rozikov and A. Zada. On dynamics of l− Volterra quadratic stochastic operators. International Journal of Biomathematics, 3(2):143–159, 2010. https://doi.org/10.1142/S1793524510000945
L.J.S. Allen. Some discrete-time SI, SIR, and SIS epidemic models. Mathematical BioSciences, 124(1):83–105, 1994. https://doi.org/10.1016/0025-5564(94)90025-6
V.M. Barashkov and N.A. Zadorina. Analysis of two-stage mathematical models of computer distribution viruses. In A.B. Yegorov, V.Ya. Hentov and at al.(Eds.), Proc. of the 5th Intern. Conference Actual Problems Technical Sciences in Russia and abroad, Novosibirsk, Russia, 2018, pp. 96–102, 2018. (in Russian)
J. Blath, U.U. Jamilov(Zhamilov) and M. Scheutzow. (G,µ)- quadratic stochastic operators. Journal of Difference Equations and Applications, 20(8):1258–1267, 2014. https://doi.org/10.1080/10236198.2014.906591
D. Chumachenko, K. Chumachenko and S. Yakovlev. Intelligent simulation of network worm propagation using the code red as an example. Telecommunications and Radio Engineering, 78(5):443–464, 2019. https://doi.org/10.1615/TelecomRadEng.v78.i5.60
D. Chumachenko and S. Yakovlev. Development of deterministic models of malicious software distribution in heterogeneous networks. In 2019 3rd International Conference on Advanced Information and Communications Technologies (AICT), pp. 439–442, 2019. https://doi.org/10.1109/AIACT.2019.8847903
Á.M. del Rey. A SIR e-epidemic model for computer worms based on cellular automata. In C. Bielza, A. Salmerón and et al. (Eds.), Advances in Artificial Intelligence, pp. 228–238, Berlin, Heidelberg, 2013. Proceedings of 15th Conference of the Spanish Association for Artificial Intelligence, CAEPIA 2013 Madrid, Spain, September 17-20, 2013.
R.L. Devaney. An introduction to chaotic dynamical systems. Studies in Nonlinearity, Westview Press, Boulder, CO, 2003.
N.N. Ganikhodjaev, R.N. Ganikhodjaev and U.U. Jamilov. Quadratic stochastic operators and zero-sum game dynamics. Ergodic Theory and Dynamical Systems, 35(5):1443–1473, 2015. https://doi.org/10.1017/etds.2013.109
N.N. Ganikhodjaev, U.U. Jamilov and R.T. Mukhitdinov. On non - ergodic transformations on S3. Journal of Physics: Conference Series, 435(1):012005, 2013. https://doi.org/10.1088/1742-6596/435/1/012005
R.N. Ganikhodzhaev. Quadratic stochastic operators, Lyapunov functions and tournaments. Russian Academy of Sciences. Sbornik Mathematics, 76(2):489– 506, 1993. https://doi.org/10.1070/SM1993v076n02ABEH003423
R.N. Ganikhodzhaev. Map of fixed points and Lyapunov functions for one class of discrete dynamical systems. Math. Notes, 56(5):1125–1131, 1994. https://doi.org/10.1007/BF02274660
R.N. Ganikhodzhaev, F.M. Mukhamedov and U.A. Rozikov. Quadratic stochastic operators and processes: results and open problems. Infinite Dimensional Analysis, Quantum Probability and Related Topics, 14(2):279–335, 2011. https://doi.org/10.1142/S0219025711004365
U. Jamilov. On a family of strictly non-Volterra quadratic stochastic operators. Journal of Physics: Conference Series, 697(1):012013, 2016. https://doi.org/10.1088/1742-6596/697/1/012013
U.U. Jamilov. Quadratic stochastic operators corresponding to graphs. Lobachevskii Journal of Mathematics, 34(2):148–151, 2013. https://doi.org/10.1134/S1995080213020042
J. O. Kephart and S. R. White. Directed-graph epidemiological models of computer viruses. In Proc. of the 1991 IEEE Computer Society Symposium on Research in Security and Privacy, Oakland, California, pp. 343–359, 1991.
H. Kesten. Quadratic transformations: A model for population growth. I. Advances in Applied Probability, 2(1):1–82, 1970. https://doi.org/10.2307/3518344
I.V. Kotenko and V.V. Vorontsov. Analytical models of network worm propagation. In SPIIRAS Proceedings, volume 4, pp. 208–224, SPb., 2007. Nauka. https://doi.org/10.15622/sp.4.15 (in Russian)
Y.I. Lyubich. Mathematical structures in population genetics. vol. 22 of Biomathematics, Springer-Verlag, Berlin, 1992.
P. Van Mieghem. The n-intertwined SIS epidemic network model. Computing, 93(2):147–169, 2011. https://doi.org/10.1007/s00607-011-0155-y
F. Mukhamedov and A.F. Embong. On stable of b-bistochastic quadratic stochastic operators and associated nonhomogenous Markov chains. Linear and Multilinear Algebra, 66(1):1–21, 2018. https://doi.org/10.1080/03081087.2017.1281215
F. Mukhamedov and N. Ganikhodjaev. Quantum Quadratic Operators and Processes. vol. 2133, Lecture Notes in Mathematics, Springer-Verlag, Berlin, 2015. https://doi.org/10.1007/978-3-319-22837-2
F. Mukhamedov, M. Saburov and I. Qaralleh. On ξ(s)-quadratic stochastic operators on two-dimensional simplex and their behavior. Abst. Appl. Anal., 2013:942038, 2013. https://doi.org/10.1155/2013/942038
U.A. Rozikov and S. K. Shoyimardonov. Ocean ecosystem discrete time dynamics generated by l− Volterra operators. International Journal of Biomathematics, 12(2):1950015, 2019. https://doi.org/10.1142/S1793524519500153
U.A. Rozikov and S.S. Xudayarov. Quadratic non-stochastic operators: examples of splitted chaos. Annals of Functional Analysis, 13(1):17, 2022. https://doi.org/10.1007/s43034-021-00162-5
U.A. Rozikov and A. Zada. On dynamics of l− Volterra quadratic stochastic operators. International Journal of Biomathematics, 3(2):143–159, 2010. https://doi.org/10.1142/S1793524510000945