Hostname: page-component-74656f9884-c52kr Total loading time: 0 Render date: 2026-11-08T21:57:34.645Z Has data issue: false hasContentIssue false
  • English
  • Français

Adversarial Deletion in a Scale-Free Random Graph Process

Published online by Cambridge University Press:  01 March 2007

ABRAHAM D. FLAXMAN ,
ALAN M. FRIEZE  and
JUAN VERA
ABRAHAM D. FLAXMAN
Affiliation:
Department of Mathematical Sciences, Carnegie Mellon UniversityPittsburgh, PA 15213-3890, USA (e-mail abie@cmu.edu, alan@random.math.cmu.edu, jvera@andrew.cmu.edu)
ALAN M. FRIEZE
Affiliation:
Department of Mathematical Sciences, Carnegie Mellon UniversityPittsburgh, PA 15213-3890, USA (e-mail abie@cmu.edu, alan@random.math.cmu.edu, jvera@andrew.cmu.edu)
JUAN VERA
Affiliation:
Department of Mathematical Sciences, Carnegie Mellon UniversityPittsburgh, PA 15213-3890, USA (e-mail abie@cmu.edu, alan@random.math.cmu.edu, jvera@andrew.cmu.edu)
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the 'Save PDF' action button.

We study a dynamically evolving random graph which adds vertices and edges using preferential attachment and is ‘attacked by an adversary’. At time t, we add a new vertex xt and m random edges incident with xt, where m is constant. The neighbours of xt are chosen with probability proportional to degree. After adding the edges, the adversary is allowed to delete vertices. The only constraint on the adversarial deletions is that the total number of vertices deleted by time n must be no larger than δn, where δ is a constant. We show that if δ is sufficiently small and m is sufficiently large then with high probability at time n the generated graph has a component of size at least n/30.

Information

Type
Paper
Information
Combinatorics, Probability and Computing , Volume 16 , Issue 2 , March 2007 , pp. 261 - 270
Copyright
Copyright © Cambridge University Press 2006

References

[1]Aiello, W., Chung, F. R. K. and Lu, L. (2000) A random graph model for massive graphs. In Proc. 32nd Annual ACM Symposium on the Theory of Computing, pp. 171–180.CrossRefGoogle Scholar
[2]Aiello, W., Chung, F. R. K. and Lu, L. (2001) Random evolution in massive graphs. In Proc. of IEEE Symposium on Foundations of Computer Science, pp. 510–519.CrossRefGoogle Scholar
[3]Albert, R., Barabási, A. and Jeong, H. (1999) Diameter of the world wide web. Nature 401 103–131,CrossRefGoogle Scholar
[4]Barabási, A. and Albert, R. (1999) Emergence of scaling in random networks. Science 286 509–512.CrossRefGoogle ScholarPubMed
[5]Berger, N., Bollobás, B., Borgs, C., Chayes, J. and Riordan, O. (2003) Degree distribution of the FKP network model. In Proc. 30th International Colloquium of Automata, Languages and Programming, pp. 725–738.CrossRefGoogle Scholar
[6]Berger, N., Borgs, C., Chayes, J., D'Souza, R. and Kleinberg, R. D. (2005) Degree distribution of competition-induced preferential attachment graphs. Combin. Probab. Comput. 14 697–721.CrossRefGoogle Scholar
[7]Bollobás, B. and Riordan, O. (2002) Mathematical results on scale-free random graphs. In Handbook of Graphs and Networks, Wiley-VCH, Berlin.Google Scholar
[8]Bollobás, B. and Riordan, O. (2003) Robustness and vulnerability of scale-free random graphs. Internet Mathematics 1 1–35.CrossRefGoogle Scholar
[9]Bollobás, B. and Riordan, O. (2004) The diameter of a scale-free random graph. Combinatorica 24 5–34.CrossRefGoogle Scholar
[10]Bollobás, B. and Riordan, O. (2004) Coupling scale-free and classical random graphs. Internet Math. 1 215–225.CrossRefGoogle Scholar
[11]Bollobás, B. and Riordan, O.Spencer, J. and Tusnády, G. (2001) The degree sequence of a scale-free random graph process. Random Struct. Alg. 18 279–290.CrossRefGoogle Scholar
[12]Broder, A., Kumar, R., Maghoul, F., Raghavan, P., Rajagopalan, S., Stata, R., Tomkins, A. and Wiener, J. (2002) Graph structure in the web. In Proc. 9th Intl. World Wide Web Conference, pp. 309–320.Google Scholar
[13]Chung, F. R. K. and Lu, L. (2004) Coupling online and offline analyses for random power law graphs. Internet Math. 1 409–461.CrossRefGoogle Scholar
[14]Chung, F. R. K., Lu, L. and Vu, V. (2003) Eigenvalues of random power law graphs. Ann. Combin. 7 21–33.CrossRefGoogle Scholar
[15]Chung, F. R. K., Lu, L. and Vu, V. (2003) The spectra of random graphs with expected degrees. Proc. Natl. Acad. Sci. 100 6313–6318.CrossRefGoogle ScholarPubMed
[16]Cooper, C. and Frieze, A. M. (2003) A general model of undirected web graphs. Random Struct. Alg. 22 311–335.CrossRefGoogle Scholar
[17]Cooper, C., Frieze, A. and Vera, J. (2004) Random deletion in a scale-free random graph process. Internet Math. 1 463–483.CrossRefGoogle Scholar
[18]Erdős, P. and Rényi, A. (1959) On random graphs I. Publ. Math. Debrecen 6 290–297.CrossRefGoogle Scholar
[19]Fabrikant, A., Koutsoupias, E. and Papadimitriou, C. H. (2002) Heuristically optimized trade-offs: A new paradigm for power laws in the internet. Proc. 29th International Colloquium of Automata, Languages and Programming.CrossRefGoogle Scholar
[20]Faloutsos, M., Faloutsos, P. and Faloutsos, C. (1999) On power-law relationships of the internet topology. In Proc. SIGCOMM'99: Proceedings of the conference on applications, technologies, architectures, and protocols for computer communication, ACM Press, pp. 251–262.CrossRefGoogle Scholar
[21]Hayes, B. (2000) Graph theory in practice: Part II. American Scientist 88 104–109.CrossRefGoogle Scholar
[22]Kleinberg, J. M., Kumar, R., Raghavan, P., Rajagopalan, S. and Tomkins, A. S. (1999) The web as a graph: Measurements, models and methods. In Proc. International Conference on Combinatorics and Computing, Vol. 1627 of Lecture Notes in Computer Science, Springer, pp. 1–17.CrossRefGoogle Scholar
[23]Kumar, R., Raghavan, P., Rajagopalan, S., Sivakumar, D., Tomkins, A. and Upfal, E. (2000) Stochastic models for the web graph, In Proc. IEEE Symposium on Foundations of Computer Science, pp. 57–65.Google Scholar
[24]McDiarmid, C. J. H. (1998) Concentration. In Probabilistic Methods in Algorithmic Discrete Mathematics, Springer, pp. 195–248.CrossRefGoogle Scholar
[25]Mihail, M. and Papadimitriou, C. H. (2002) On the eigenvalue power law. In Proc. 6th International Workshop on Randomization and Approximation Techniques, pp. 254–262.CrossRefGoogle Scholar
[26]Mihail, M., Papadimitriou, C. H. and Saberi, A. (2006) On certain connectivity properties of the internet topology. J. Comput. System Sci. 72 239–251.CrossRefGoogle Scholar
[27]Mitzenmacher, M. (2004) A brief history of generative models for power law and lognormal distributions. Internet Math. 1 226–251.CrossRefGoogle Scholar
[28]Penrose, M. D. (2003) Random Geometric Graphs, Oxford University Press.CrossRefGoogle Scholar
[29]Simon, H. A. (1955) On a class of skew distribution functions. Biometrika 42 425–440.CrossRefGoogle Scholar
[30]Watts, D. J. (1999) Small Worlds: The Dynamics of Networks between Order and Randomness, Princeton University Press.CrossRefGoogle Scholar
[31]Yule, G. (1925) A mathematical theory of evolution based on the conclusions of Dr. J. C. Willis. Philos. Trans. Royal Soc. London, Ser. B 213 21–87.Google Scholar