Abstract
We obtain tight thresholds for bond percolation on one-dimensional small-world graphs, and apply such results to obtain tight thresholds for the Independent Cascade process and the Reed-Frost process in such graphs.
Although one-dimensional small-world graphs are an idealized and unrealistic network model, a number of realistic qualitative epidemiological phenomena emerge from our analysis, including the epidemic spread through a sequence of local outbreaks, the danger posed by random connections, and the effect of super-spreader events.
LT’s work on this project has received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (grant agreement No. 834861). LB’s work on this project was partially supported by the ERC Advanced Grant 788893 AMDROMA, the EC H2020RIA project “SoBigData++” (871042), the MIUR PRIN project ALGADIMAR. AC’s and FP’s work on this project was partially supported by the University of Rome “Tor Vergata” under research program “Beyond Borders” project ALBLOTECH (grant no. E89C20000620005).
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Notes
- 1.
The main difference is that Information Cascade allows the probability of “transmission” along an edge (u, v) to be a quantity \(p_{(u,v)}\), but this generalization would also make sense and be well defined in the Reed-Frost model and in the percolation process. The case in which all the probabilities are equal is called the homogenous case.
- 2.
We recall that the 3-\(\mathcal {SWG}(n)\) model and random 3-regular graphs are contiguous, i.e. each property that holds with probability \(1-o(1)\) on one of the two models, holds with probability \(1-o(1)\) also in the other one [21].
- 3.
As usual, we say that an event E holds with high probability (for short, w.h.p.) if a constant \(\gamma >0\) exists such that \(\textbf{Pr} \left( E \right) >1-n^{-\gamma }\).
- 4.
The quantity \(R_0\) in a SIR process is the expected number of people that an infectious person transmits the infection to, if all the contacts of that person are susceptible. In the percolation view of the process, it is the average degree of the percolation graph \(G_p\).
- 5.
- 6.
With respect to \(\textrm{SIR}\), for each node we have a fourth, Exposed state, corresponding to the incubation period of a node.
- 7.
We state the result for the case \(|I_0|=1\).
References
Alon, N., Benjamini, I., Stacey, A.: Percolation on finite graphs and isoperimetric inequalities. Ann. Probab. 32(3), 1727–1745 (2004)
Alon, N., Spencer, J.H.: The Probabilistic Method. Wiley Publishing, 2nd edn. (2000)
Barabási, A.L., Albert, R.: Emergence of scaling in random networks. Science 286(5439), 509–512 (1999). https://doi.org/10.1126/science.286.5439.509
Becchetti, L., Clementi, A., Denni, R., Pasquale, F., Trevisan, L., Ziccardi, I.: Percolation and epidemic processes in one-dimensional small-world networks. arXiv e-prints pp. arXiv-2103 (2021)
Becchetti, L., Clementi, A., Pasquale, F., Trevisan, L., Ziccardi, I.: Bond percolation in small-world graphs with power-law distribution. arXiv preprint arXiv:2205.08774 (2022)
Benjamini, I., Berger, N.: The diameter of long-range percolation clusters on finite cycles. Random Struct. Algorithms 19(2), 102–111 (2001). https://doi.org/10.1002/rsa.1022
Biskup, M.: On the scaling of the chemical distance in long-range percolation models. Ann. Probability 32(4), 2938–2977 (2004). https://doi.org/10.1214/009117904000000577
Biskup, M.: Graph diameter in long-range percolation. Random Struct. Algorithms 39(2), 210–227 (2011). https://doi.org/10.1002/rsa.20349
Bollobás, B., Janson, S., Riordan, O.: The phase transition in inhomogeneous random graphs. Random Struct. Algorithms 31(1), 3–122 (2007). https://doi.org/10.1002/rsa.20168. https://onlinelibrary.wiley.com/doi/abs/10.1002/rsa.20168
Bollobás, B., Riordan, O.: The diameter of a scale-free random graph. Comb. 24(1), 5–34 (2004). https://doi.org/10.1007/s00493-004-0002-2
Callaway, D.S., Hopcroft, J.E., Kleinberg, J.M., Newman, M.E.J., Strogatz, S.H.: Are randomly grown graphs really random? Phys. Rev. E 64, 041902 (2001). https://doi.org/10.1103/PhysRevE.64.041902
Chen, W., Lakshmanan, L., Castillo, C.: Information and influence propagation in social networks. Synthesis Lectures Data Manage. 5, 1–177 (10 2013). https://doi.org/10.2200/S00527ED1V01Y201308DTM037
Chen, W., Lu, W., Zhang, N.: Time-critical influence maximization in social networks with time-delayed diffusion process. In: Proceedings of the Twenty-Sixth AAAI Conference on Artificial Intelligence, AAAI 2012, pp. 592–598. AAAI Press (2012)
Choi, H., Pant, M., Guha, S., Englund, D.: Percolation-based architecture for cluster state creation using photon-mediated entanglement between atomic memories. NPJ Quantum Information 5(1), 104 (2019)
Draief, M., Ganesh, A., Massoulié, L.: Thresholds for virus spread on networks. In: Proceedings of the 1st International Conference on Performance Evaluation Methodolgies and Tools, p. 51-es. valuetools 2006. Association for Computing Machinery, New York (2006). https://doi.org/10.1145/1190095.1190160
Durrett, R., Kesten, H.: The critical parameter for connectedness of some random graphs. A Tribute to P. Erdos, pp. 161–176 (1990)
Easley, D., Kleinberg, J.: Networks, Crowds, and Markets: Reasoning About a Highly Connected World. Cambridge University Press, Cambridge (2010)
Garetto, M., Gong, W., Towsley, D.: Modeling malware spreading dynamics. In: IEEE INFOCOM 2003. Twenty-second Annual Joint Conference of the IEEE Computer and Communications Societies (IEEE Cat. No.03CH37428), vol. 3, pp. 1869–1879 (2003). https://doi.org/10.1109/INFCOM.2003.1209209
Goerdt, A.: The giant component threshold for random regular graphs with edge faults. Theor. Comput. Sci. 259(1-2), 307–321 (2001). https://doi.org/10.1016/S0304-3975(00)00015-3
Janson, S.: Large deviations for sums of partly dependent random variables. Random Struct. Algorithms 24, May 2004. https://doi.org/10.1002/rsa.20008
Janson, S., Rucinski, A., Luczak, T.: Random Graphs. Wiley, New York (2011)
Karlin, A.R., Nelson, G., Tamaki, H.: On the fault tolerance of the butterfly. In: Proceedings of the Twenty-Sixth Annual ACM Symposium on Theory of Computing, pp. 125–133 (1994)
Kempe, D., Kleinberg, J., Tardos, E.: Maximizing the spread of influence through a social network. Theory Comput. 11(4), 105–147 (2015). https://doi.org/10.4086/toc.2015.v011a004, (An extended abstract appeared in Proc. of 9th ACM KDD ’03)
Kesten, H.: The critical probability of bond percolation on the square lattice equals 1/2. Commun. Math. Phys. 74(1), 41–59 (1980)
Kott, A., Linkov, I. (eds.): Cyber Resilience of Systems and Networks. RSD, Springer, Cham (2019). https://doi.org/10.1007/978-3-319-77492-3
Lee, E.J., Kamath, S., Abbe, E., Kulkarni, S.R.: Spectral bounds for independent cascade model with sensitive edges. In: 2016 Annual Conference on Information Science and Systems, CISS 2016, Princeton, NJ, USA, 16–18 March 2016, pp. 649–653. IEEE (2016). https://doi.org/10.1109/CISS.2016.7460579
Lemonnier, R., Seaman, K., Vayatis, N.: 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 2014, pp. 846–854. MIT Press, Cambridge (2014)
Lin, Q., et al.: A conceptual model for the coronavirus disease 2019 (COVID-19) outbreak in Wuhan, China with individual reaction and governmental action. Int. J. Infect. Dis. 93, 211–216 (2020)
Liu, B., Cong, G., Xu, D., Zeng, Y.: Time constrained influence maximization in social networks. In: Proceedings of the IEEE International Conference on Data Mining, ICDM, pp. 439–448, December 2012. https://doi.org/10.1109/ICDM.2012.158
Moore, C., Newman, M.E.: Exact solution of site and bond percolation on small-world networks. Phys. Rev. E 62, 7059–7064 (2000). https://doi.org/10.1103/PhysRevE.62.7059
Moore, C., Newman, M.E.: Epidemics and percolation in small-world networks. Phys. Rev. E 61, 5678–5682 (2000). https://doi.org/10.1103/PhysRevE.61.5678
Newman, C.M., Schulman, L.S.: One dimensional \(1/| j- i|^s\) percolation models: the existence of a transition for \(s \le 2\). Commun. Math. Phys. 104(4), 547–571 (1986)
Newman, M.E., Watts, D.J.: Scaling and percolation in the small-world network model. Phys. Rev. E 60(6), 7332 (1999)
Pastor-Satorras, R., Castellano, C., Van Mieghem, P., Vespignani, A.: Epidemic processes in complex networks. Rev. Mod. Phys. 87, 925–979 (2015). https://doi.org/10.1103/RevModPhys.87.925
Shante, V.K., Kirkpatrick, S.: An introduction to percolation theory. Adv. Phys. 20(85), 325–357 (1971). https://doi.org/10.1080/00018737100101261
Watts, D.J., Strogatz, S.H.: Collective dynamics of ‘small-world’ networks. Nature 393(6684), 440–442 (1998)
Wei, W., Ming, T., Eugene, S., Braunstein, L.A.: Unification of theoretical approaches for epidemic spreading on complex networks. Reports Progress Phys. 80(3), 036603 (2017). https://doi.org/10.1088/1361-6633/aa5398
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2022 Springer Nature Switzerland AG
About this paper
Cite this paper
Becchetti, L., Clementi, A., Denni, R., Pasquale, F., Trevisan, L., Ziccardi, I. (2022). Percolation and Epidemic Processes in One-Dimensional Small-World Networks. In: Castañeda, A., Rodríguez-Henríquez, F. (eds) LATIN 2022: Theoretical Informatics. LATIN 2022. Lecture Notes in Computer Science, vol 13568. Springer, Cham. https://doi.org/10.1007/978-3-031-20624-5_29
Download citation
DOI: https://doi.org/10.1007/978-3-031-20624-5_29
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-031-20623-8
Online ISBN: 978-3-031-20624-5
eBook Packages: Computer ScienceComputer Science (R0)