Bond Percolation in Small-World Graphs with Power-Law Distribution
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Bond Percolation in Small-World Graphs with Power-Law Distribution

Authors Luca Becchetti, Andrea Clementi, Francesco Pasquale, Luca Trevisan, Isabella Ziccardi

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Author Details

Luca Becchetti
  • Sapienza University of Rome, Italy
Andrea Clementi
  • University of Rome Tor Vergata, Italy
Francesco Pasquale
  • University of Rome Tor Vergata, Italy
Luca Trevisan
  • Bocconi University, Milan, Italy
Isabella Ziccardi
  • Bocconi University, Milan, Italy

Funding

Luca Trevisan and Isabella Ziccardi: Supported by the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (grant agreement No. 834861).
  • Becchetti, Luca: Supported by the ERC Advanced Grant 788893 AMDROMA, EC H2020RIA project SoBigData++ (871042), PNRR MUR project PE0000013-FAIR, PNRR MUR project IR0000013-SoBigData.it.
  • Clementi, Andrea: Partially supported by Spoke 1 “FutureHPC & BigData” of the Italian Research Center on High-Performance Computing, Big Data and Quantum Computing (ICSC) funded by MUR Missione 4 Componente 2 Investimento 1.4: Potenziamento strutture di ricerca e creazione di “campioni nazionali” di R&S (M4C2-19) - Next Generation EU (NGEU).

Cite As Get BibTex

Luca Becchetti, Andrea Clementi, Francesco Pasquale, Luca Trevisan, and Isabella Ziccardi. Bond Percolation in Small-World Graphs with Power-Law Distribution. In 2nd Symposium on Algorithmic Foundations of Dynamic Networks (SAND 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 257, pp. 3:1-3:22, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023) https://doi.org/10.4230/LIPIcs.SAND.2023.3

Abstract

Full-bond percolation with parameter p is the process in which, given a graph, for every edge independently, we keep the edge with probability p and delete it with probability 1-p. Bond percolation is studied in parallel computing and network science to understand the resilience of distributed systems to random link failure and the spread of information in networks through unreliable links. Moreover, the full-bond percolation is equivalent to the Reed-Frost process, a network version of SIR epidemic spreading.
We consider one-dimensional power-law small-world graphs with parameter α obtained as the union of a cycle with additional long-range random edges: each pair of nodes {u,v} at distance L on the cycle is connected by a long-range edge {u,v}, with probability proportional to 1/L^α. Our analysis determines three phases for the percolation subgraph G_p of the small-world graph, depending on the value of α.
- If α < 1, there is a p < 1 such that, with high probability, there are Ω(n) nodes that are reachable in G_p from one another in 𝒪(log n) hops; 
- If 1 < α < 2, there is a p < 1 such that, with high probability, there are Ω(n) nodes that are reachable in G_p from one another in log^{𝒪(1)}(n) hops; 
- If α > 2, for every p < 1, with high probability all connected components of G_p have size 𝒪(log n).

Subject Classification

ACM Subject Classification
  • Theory of computation → Random network models
Keywords
  • Information spreading
  • gossiping
  • epidemics
  • fault-tolerance
  • network self-organization and formation
  • complex systems
  • social and transportation networks

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References

  1. Noga Alon, Benjamini Itai, and Stacey Alan. Percolation on finite graphs and isoperimetric inequalities. Annals of Probability, 32:1727-1745, 2004. Google Scholar
  2. Luca Becchetti, Andrea Clementi, Francesco Pasquale, Luca Trevisan, and Isabella Ziccardi. Bond percolation in small-world graphs with power-law distribution. arXiv preprint, 2022. URL: https://arxiv.org/abs/2205.08774.
  3. Luca Becchetti, Andrea E. F. Clementi, Riccardo Denni, Francesco Pasquale, Luca Trevisan, and Isabella Ziccardi. Percolation and epidemic processes in one-dimensional small-world networks - (extended abstract). In Proceedings of LATIN 2022: the 15th Latin American Symposium, volume 13568 of Lecture Notes in Computer Science, pages 476-492. Springer, 2022. URL: https://doi.org/10.1007/978-3-031-20624-5_29.
  4. Itai Benjamini and Noam Berger. The diameter of long-range percolation clusters on finite cycles. Random Struct. Algorithms, 19(2):102-111, 2001. URL: https://doi.org/10.1002/rsa.1022.
  5. Marek Biskup. On the scaling of the chemical distance in long-range percolation models. The Annals of Probability, 32(4):2938-2977, 2004. URL: https://doi.org/10.1214/009117904000000577.
  6. Marek Biskup. Graph diameter in long-range percolation. Random Structures & Algorithms, 39(2):210-227, 2011. URL: https://doi.org/10.1002/rsa.20349.
  7. Duncan S. Callaway, John E. Hopcroft, Jon Kleinberg, Mark E. J. Newman, and Steven H. Strogatz. Are randomly grown graphs really random? Phys. Rev. E, 64:041902, September 2001. URL: https://doi.org/10.1103/PhysRevE.64.041902.
  8. Wei Chen, Laks Lakshmanan, and Carlos Castillo. Information and influence propagation in social networks. Synthesis Lectures on Data Management, 5:1-177, October 2013. URL: https://doi.org/10.2200/S00527ED1V01Y201308DTM037.
  9. Hyeongrak Choi, Mihir Pant, Saikat Guha, and Dirk Englund. Percolation-based architecture for cluster state creation using photon-mediated entanglement between atomic memories. npj Quantum Information, 5(1):104, 2019. Google Scholar
  10. Easley David and Kleinberg Jon. Networks, Crowds, and Markets: Reasoning About a Highly Connected World. Cambridge University Press, USA, 2010. Google Scholar
  11. Michele Garetto, Weibo Gong, and Donald F. Towsley. Modeling malware spreading dynamics. In Proceedings IEEE INFOCOM 2003, The 22nd Annual Joint Conference of the IEEE Computer and Communications Societies, San Franciso, CA, USA, March 30 - April 3, 2003, pages 1869-1879. IEEE Computer Society, 2003. URL: https://doi.org/10.1109/INFCOM.2003.1209209.
  12. Anna R. Karlin, Greg Nelson, and Hisao Tamaki. On the fault tolerance of the butterfly. In Proceedings of the twenty-sixth annual ACM symposium on Theory of Computing, pages 125-133, 1994. Google Scholar
  13. David Kempe, Jon Kleinberg, and Éva Tardos. Maximizing the spread of influence through a social network. Theory of Computing, 11(4):105-147, 2015. URL: https://doi.org/10.4086/toc.2015.v011a004.
  14. Harry Kesten. The critical probability of bond percolation on the square lattice equals 1/2. Communications in mathematical physics, 74(1):41-59, 1980. Google Scholar
  15. Jon Kleinberg. Navigation in a small world. Nature, 406:845, 2000. Google Scholar
  16. Alexander Kott and Igor Linkov. Cyber resilience of systems and networks. Springer, 2019. Google Scholar
  17. Rémi Lemonnier, Kevin Seaman, and Nicolas Vayatis. Tight bounds for influence in diffusion networks and application to bond percolation and epidemiology. In Proceedings of the 27th International Conference on Neural Information Processing Systems - Volume 1, NIPS'14, pages 846-854, Cambridge, MA, USA, 2014. MIT Press. Google Scholar
  18. Cristopher Moore and Mark E. J. Newman. Epidemics and percolation in small-world networks. Physical Review E, 61(5):5678, 2000. Google Scholar
  19. Cristopher Moore and Mark E. J. Newman. Exact solution of site and bond percolation on small-world networks. Phys. Rev. E, 62:7059-7064, November 2000. URL: https://doi.org/10.1103/PhysRevE.62.7059.
  20. Charles M. Newman and Lawrence S. Schulman. One dimensional 1/|j-i|^s percolation models: The existence of a transition for s ≦ 2. Communications in Mathematical Physics, 104(4):547-571, 1986. Google Scholar
  21. Mark E. J. Newman and Duncan J. Watts. Scaling and percolation in the small-world network model. Physical review E, 60(6):7332, 1999. Google Scholar
  22. Romualdo Pastor-Satorras, Claudio Castellano, Piet Van Mieghem, and Alessandro Vespignani. Epidemic processes in complex networks. Rev. Mod. Phys., 87:925-979, August 2015. URL: https://doi.org/10.1103/RevModPhys.87.925.
  23. Vinod K.S. Shante and Scott Kirkpatrick. An introduction to percolation theory. Advances in Physics, 20(85):325-357, 1971. URL: https://doi.org/10.1080/00018737100101261.
  24. Wei Wang, Ming Tang, H Eugene Stanley, and Lidia A Braunstein. Unification of theoretical approaches for epidemic spreading on complex networks. Reports on Progress in Physics, 80(3):036603, February 2017. URL: https://doi.org/10.1088/1361-6633/aa5398.
  25. Duncan J. Watts and Steven H. Strogatz. Collective dynamics of "small-world" networks. Nature, 393(6684):440-442, 1998. Google Scholar
  26. Tongfeng Weng, Michael Small, Jie Zhang, and Pan Hui. Lévy walk navigation in complex networks: A distinct relation between optimal transport exponent and network dimension. Scientific Reports, 5(1):17309, 2015. Google Scholar
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