Abstract
We study vulnerability of a uniformly distributed random graph to an attack by an adversary who aims for a global change of the distribution while being able to make only a local change in the graph. We call a graph property A anti-stochastic if the probability that a random graph G satisfies A is small but, with high probability, there is a small perturbation transforming G into a graph satisfying A. While for labeled graphs such properties are easy to obtain from binary covering codes, the existence of anti-stochastic properties for unlabeled graphs is not so evident. If an admissible perturbation is either the addition or the deletion of one edge, we exhibit an anti-stochastic property that is satisfied by a random unlabeled graph of order n with probability \((2+o(1))/n^2\), which is as small as possible. We also express another anti-stochastic property in terms of the degree sequence of a graph. This property has probability \((2+o(1))/(n\ln n)\), which is optimal up to factor of 2.
The third author is supported by DFG grant KO 1053/8–2.
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References
Babai, L., Erdős, P., Selkow, S.M.: Random graph isomorphism. SIAM J. Comput. 9(3), 628–635 (1980). https://doi.org/10.1137/0209047
Bollobás, B.: Random Graphs. Cambridge Studies in Advanced Mathematics, vol. 73, 2nd edn. Cambridge University Press (2001). https://doi.org/10.1017/CBO9780511814068
Bollobás, B., Riordan, O.: Coupling scale-free and classical random graphs. Internet Math. 1(2), 215–225 (2003). https://doi.org/10.1080/15427951.2004.10129084
Bollobás, B., Riordan, O.: Robustness and vulnerability of scale-free random graphs. Internet Math. 1(1), 1–35 (2003). https://doi.org/10.1080/15427951.2004.10129080
Cohen, G.D., Honkala, I.S., Litsyn, S., Lobstein, A.: Covering Codes. North-Holland Mathematical Library, vol. 54. North-Holland (2005)
Firer, M.: Alternative metrics. In: Concise Encyclopedia of Coding Theory, pp. 555–574. CRC Press (2021)
Flaxman, A.D., Frieze, A.M., Vera, J.: Adversarial deletion in a scale-free random graph process. Comb. Probab. Comput. 16(2), 261–270 (2007). https://doi.org/10.1017/S0963548306007681
Harary, F., Palmer, E.M.: Graphical Enumeration. Academic Press, New York-London (1973)
Herschkorn, S.J.: On the modular value and fractional part of a random variable. Probab. Eng. Inf. Sci. 9(4), 551–562 (1995). https://doi.org/10.1017/S0269964800004058
Janson, S., Łuczak, T., Ruciński, A.: Random Graphs. Wiley, Hoboken (2000)
Kabatyanskiĭ, G.A., Panchenko, V.I.: Packings and coverings of the Hamming space by balls of unit radius. Problems Inform. Transm. 24(4), 261–272 (1988)
Kiselev, S., Kupavskii, A., Verbitsky, O., Zhukovskii, M.: On anti-stochastic properties of unlabeled graphs. arXiv:2112.04395 (2021)
Liebenau, A., Wormald, N.: Asymptotic enumeration of graphs by degree sequence, and the degree sequence of a random graph. arXiv:1702.08373 (2018)
McKay, B.D., Wormald, N.C.: The degree sequence of a random graph I. The models. Random Struct. Algorithms 11(2), 97–117 (1997)
Schaeffer, S., Valdés, V., Figols, J., Bachmann, I., Morales, F., Bustos-Jiménez, J.: Characterization of robustness and resilience in graphs: a mini-review. J. Complex Netw. 9(2) (2021). https://doi.org/10.1093/comnet/cnab018
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Kiselev, S., Kupavskii, A., Verbitsky, O., Zhukovskii, M. (2022). On Anti-stochastic Properties of Unlabeled Graphs. In: Bekos, M.A., Kaufmann, M. (eds) Graph-Theoretic Concepts in Computer Science. WG 2022. Lecture Notes in Computer Science, vol 13453. Springer, Cham. https://doi.org/10.1007/978-3-031-15914-5_22
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