Modeling Cyber-Security

  • Nicholas J. DarasEmail author
  • Argyrios Alexopoulos
Part of the Springer Optimization and Its Applications book series (SOIA, volume 131)


This paper documents a holistic mathematical modeling theory to provide a rigorous description of cyber-attacks and cyber-security. After determining valuations and vulnerabilities of parts of a node constituent, we recall the definitions of cyber-effect and cyber-interaction. Based on these concepts, we give the mathematical definitions of cyber navigation and infected node and we explain what is meant by dangerous cyber navigation and protection of cyber nodes from unplanned attacks. Our discussion proceeds to a rigorous description of passive and active cyber-attacks, as well as the relevant protections.


Mathematical modeling (models of systems) Internet topics Measure theory Complex spaces Valuation of a part of node constituent Vulnerability of a part of node constituent Node supervision Cyber-effect Cyber-interaction Germ of cyber-attack Cyber defense Proactive cyber protection 


  1. 1.
    Adams, N., Heard, N. (eds.): Data Analysis for Network Cyber-Security, 200 pp. World Scientific, Singapore (2014). ISBN: 978-1-78326-374-5 (hardcover)/ISBN: 978-1-78326-376-9 (ebook)Google Scholar
  2. 2.
    Alexopoulos, A., Daras, N.: Case study of proactive defense against various types of cyber attacks, submittedGoogle Scholar
  3. 3.
    Colbaugh, R., Glass, K.: Proactive Cyber Defense. Springer Integrated Series on Intelligent Systems. Springer, Berlin (2012). Document No. 5299122, SAND 2011-8794PGoogle Scholar
  4. 4.
    Daras, N.J.: On the mathematical definition of cyberspace. 4th International Conference on Operational Planning, Technological Innovations and Mathematical Applications (OPTIMA), Hellenic Army Academy, Vari Attikis, Greece (2017)Google Scholar
  5. 5.
    Daras, N., Alexopoulos, A.: Mathematical description of cyber-attacks and proactive defenses. Journal of Applied Mathematics & Bioinformatics, 7(1), 71–142 (2017)Google Scholar
  6. 6.
    Dunlavy, D.M., Hendrickson, B., Kolda, T.G.: Mathematical challenges in cyber-security. Sandia National Laboratories (2009).
  7. 7.
    Luccio, F., Pagli, L., Stee, G.: Mathematical and Algorithmic Foundations of the Internet, pp. 221. CRC, Boca Raton (2011)Google Scholar
  8. 8.
    Malkevitch, J.: Mathematics and internet security (2015).
  9. 9.
    Meza, J., Campbell, S., Bailey, D.: Mathematical and statistical opportunities in cyber security (2009).
  10. 10.
    Ramsey, D.: Mathematical problems in cyber security. In: DOE Workshop on Mathematical Research Challenges in Optimization of Complex Systems, 7–8 December 2006.
  11. 11.
    Rowe, N.C., Custy E.J., Duong B.T.: Defending cyberspace with fake honeypots. J. Comput. 2(2), 25–36 (2007). CrossRefGoogle Scholar
  12. 12.
    Ryan, P.Y.A.: Mathematical models of computer security. In: Focardi, R., Gorrieri, R. (eds.) Foundations of Security Analysis and Design: Tutorial Lectures (FOSAD 2000). Lecture Notes in Computer Science, vol. 2171, pp. 1–62. Springer, Berlin, Heidelberg (2001). Google Scholar
  13. 13.
    Saini, D.K.: Cyber defense: mathematical modeling and simulation. Int. J. Appl. Phys. Math. 2(5), 312–315 (2012)CrossRefGoogle Scholar
  14. 14.
    Shen, D., Chen, G., Cruz, J.B. Jr., Blasch, E., Pham, K.: An adaptive Markov game model for cyber threat intent inference. In: Theory and Novel Applications of Machine Learning, pp. 376. I-Tech, Vienna (2009). ISBN 978-3-902613-55-4.
  15. 15.
    Subil, A., Suku, N.: Cyber security analytics: a stochastic model for security quantification using absorbing Markov chains. J. Commun. 9(12), 899–907 (2014)Google Scholar
  16. 16.
    Wendelberger, J., Griffin, C., Wilder, L., Jiao, Y., Kolda T.: A mathematical basis for science-based cyber-security (with alternate title “Mathematical Underpinnings for Science-Based Cyber-Security”), February 2008, Sandia National Laboratories, Chad Scherrer, Pacific Northwest National Laboratory.

Copyright information

© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of MathematicsHellenic Military AcademyVari AttikisGreece
  2. 2.Croisy Sur SeineFrance

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