On the Complexity of Color-Avoiding Site and Bond Percolation
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.
KeywordsComputational complexity Color-avoiding percolation Robustly connected components Attack tolerance Shared vulnerability
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.
- 9.Hackett, A., Cellai, D., Gómez, S., Arenas, A., Gleeson, J.P.: Bond percolation on multiplex networks. Phys. Rev. X 6(2), 021002 (2016)Google Scholar
- 10.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)
- 11.Kadović, A., Zlatić, V.: Color-avoiding edge percolation on edge-colored network. In: Complenet (2017)Google Scholar
- 12.Krause, S.M., Danziger, M.M., Zlatić, V.: Hidden connectivity in networks with vulnerable classes of nodes. Phys. Rev. X 6(4), 041022 (2016)Google Scholar