Abstract
The mathematical analysis of robustness and error-tolerance of complex networks has been in the center of research interest. On the other hand, little work has been done when the attack-tolerance of the vertices or edges are not independent but certain classes of vertices or edges share a mutual vulnerability. In this study, we consider a graph and we assign colors to the vertices or edges, where the color-classes correspond to the shared vulnerabilities. An important problem is to find robustly connected vertex sets: nodes that remain connected to each other by paths providing any type of error (i.e. erasing any vertices or edges of the given color). This is also known as color-avoiding percolation.
In this paper, we study various possible modeling approaches of shared vulnerabilities, we analyze the computational complexity of finding the robustly (color-avoiding) connected components. We find that the presented approaches differ significantly regarding their complexity.
This is a preview of subscription content, log in via an institution.
Buying options
Tax calculation will be finalised at checkout
Purchases are for personal use only
Learn about institutional subscriptionsReferences
Albert, R., Jeong, H., Barabási, A.-L.: Error and attack tolerance of complex networks. Nature 406(6794), 378 (2000)
Barabási, A.-L., et al.: Network Science. Cambridge University Press, Cambridge (2016)
Callaway, D.S., Newman, M.E.J., Strogatz, S.H., Watts, D.J.: Network robustness and fragility: percolation on random graphs. Phys. Rev. Lett. 85(25), 5468 (2000)
Dondi, R., Sikora, F.: Finding disjoint paths on edge-colored graphs: more tractability results. J. Comb. Optim. 36(4), 1315–1332 (2018)
Dorogovtsev, S.N., Goltsev, A.V., Mendes, J.F.F.: Kcore organization of complex networks. Phys. Rev. Lett. 96(4), 040601 (2006)
Gavril, F.: Intersection graphs of Helly families of subtrees. Discrete Appl. Math. 66(1), 45–56 (1996)
Gourves, L., Lyra, A., Martinhon, C.A., Monnot, J.: On paths, trails and closed trails in edge-colored graphs. Discrete Math. Theor. Comput. Sci. 14(2), 57–74 (2012)
Granata, D., Behdani, B., Pardalos, P.M.: On the complexity of path problems in properly colored directed graphs. J. Combin. Optim. 24(4), 459–467 (2012)
Hackett, A., Cellai, D., Gómez, S., Arenas, A., Gleeson, J.P.: Bond percolation on multiplex networks. Phys. Rev. X 6(2), 021002 (2016)
Kadović, A., Krause, S.M., Caldarelli, G., Zlatic, V.: Bond and site color-avoiding percolation in scale free networks. arXiv preprint arXiv:1807.08553 (2018)
Kadović, A., Zlatić, V.: Color-avoiding edge percolation on edge-colored network. In: Complenet (2017)
Krause, S.M., Danziger, M.M., Zlatić, V.: Hidden connectivity in networks with vulnerable classes of nodes. Phys. Rev. X 6(4), 041022 (2016)
Krause, S.M., Danziger, M.M., Zlatić, V.: Color-avoiding percolation. Phys. Rev. E 96(2), 022313 (2017)
Malaguti, E., Toth, P.: A survey on vertex coloring problems. Int. Trans. Oper. Res. 17(1), 1–34 (2010)
Newman, M.: Networks. Oxford University Press, Oxford (2018)
Penrose, M.D.: On k-connectivity for a geometric random graph. Random Struct. Algorithms 15(2), 145–164 (1999)
Santos, R.F., Andrioni, A., Drummond, A.C., Xavier, E.C.: Multicolour paths in graphs: NP-hardness, algorithms, and applications on routing in WDM networks. J. Combin. Optim. 33(2), 742–778 (2017)
Shao, S., Huang, X., Stanley, H.E., Havlin, S.: Percolation of localized attack on complex networks. New J. Phys. 17(2), 023049 (2015)
Shekhtman, L.M., et al.: Critical field-exponents for secure message-passing in modular networks. New J. Phys. 20(5), 053001 (2018)
Siganos, G., Faloutsos, M., Faloutsos, P., Faloutsos, C.: Power laws and the AS-level internet topology. IEEE/ACM Trans. Netw. (TON) 11(4), 514–524 (2003)
Sipser, M.: Introduction to the Theory of Computation. Cengage Learning, Boston (2012)
Son, S.-W., Bizhani, G., Christensen, C., Grassberger, P., Paczuski, M.: Percolation theory on interdependent networks based on epidemic spreading. EPL (Europhys. Lett.) 97(1), 16006 (2012)
Stauffer, D., Aharony, A.: Introduction to Percolation Theory: Revised Second Edition. CRC Press, Boca Raton (2014)
Wu, B.Y.: On the maximum disjoint paths problem on edge-colored graphs. Discrete Optim. 9(1), 50–57 (2012)
Yuan, X., Dai, Y., Stanley, H.E., Havlin, S.: k-core percolation on complex networks: comparing random, localized, and targeted attacks. Phys. Rev. E 93(6), 062302 (2016)
Zhao, L., Park, K., Lai, Y.-C., Ye, N.: Tolerance of scale-free networks against attack-induced cascades. Phys. Rev. E 72(2), 025104 (2005)
Acknowledgment
We thank Michael Danziger, Panna Fekete and Balázs Ráth for useful conversations. The research reported in this paper was supported by the BME- Artificial Intelligence FIKP grant of EMMI (BME FIKP-MI/SC). The publication is also supported by the EFOP-3.6.2-16-2017-00015 project entitled “Deepening the activities of HU-MATHS-IN, the Hungarian Service Network for Mathematics in Industry and Innovations” through University of Debrecen. The work of both authors is partially supported by the NKFI FK 123962 grant. R. M. is supported by NKFIH K123782 grant and by MTA-BME Stochastics Research Group.
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2019 Springer Nature Switzerland AG
About this paper
Cite this paper
Molontay, R., Varga, K. (2019). On the Complexity of Color-Avoiding Site and Bond Percolation. In: Catania, B., Královič, R., Nawrocki, J., Pighizzini, G. (eds) SOFSEM 2019: Theory and Practice of Computer Science. SOFSEM 2019. Lecture Notes in Computer Science(), vol 11376. Springer, Cham. https://doi.org/10.1007/978-3-030-10801-4_28
Download citation
DOI: https://doi.org/10.1007/978-3-030-10801-4_28
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-030-10800-7
Online ISBN: 978-3-030-10801-4
eBook Packages: Computer ScienceComputer Science (R0)