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Mathematical Models and Methods for Planet Earth pp 157–170Cite as

Superprocesses as Models for Information Dissemination in the Future Internet

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Part of the book series: Springer INdAM Series ((SINDAMS,volume 6))

Abstract

Future Internet will be composed by a tremendous number of potentially interconnected people and devices, offering a variety of services, applications and communication opportunities. In particular, short-range wireless communications, which are available on almost all portable devices, will enable the formation of the largest cloud of interconnected, smart computing devices mankind has ever dreamed about: the Proximate Internet. In this paper,we consider superprocesses, more specifically super Brownian motion, as a suitable mathematical model to analyse a basic problem of information dissemination arising in the context of Proximate Internet. The proposed model provides a promising analytical framework to both study theoretical properties related to the information dissemination process and to devise efficient and reliable simulation schemes for very large systems.

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Notes

  1. 1.

    1 Clean Slate Program, Stanford University, http://cleanslate.stanford.edu..

  2. 2.

    2 CRASH — Clean-slate Resilient Adaptive Secure Hosts, Cornell University, http://www.nuprl.org/.

References

  1. Alizadeh, M., Javanmard, A., Prabhakar, B.: Analysis of DCTCP: stability, convergence, and fairness. In: Proceedings of the ACM SIGMETRICS joint international conference on Measurement and modeling of computer systems, pp. 73–84 (2011)

    Google Scholar 

  2. Athreya, K.B., Jagers: Classical and modern branching processes. Springer, New York (1997)

    Book  Google Scholar 

  3. Athreya, K.B., Ney, P.E.: Branching processes, vol. 28. Springer, New York (1972)

    Book  Google Scholar 

  4. Barabási, A-L., Albert, R.: Emergence of scaling in random networks. Science {pb286}(5439), 509–512 (1999)

    Google Scholar 

  5. Bertoin, R.: Lévy processes, vol. 121. Cambridge University Press, Cambridge (1998)

    Google Scholar 

  6. Bingham, N.H.: Continuous branching processes and spectral positivity. Stochastic Processes and their Applications 4(3), 217–242 (1976)

    Article  Google Scholar 

  7. Blumenthal, Getoor, R.K.: Markov processes and potential theory. Courier Dover Publications, New York (1968)

    Google Scholar 

  8. Brin, S., Page, L.: The anatomy of a large-scale hypertextual Web search engine. Computer networks and ISDN systems 30(1), 107–117 (1998)

    Article  Google Scholar 

  9. Brockmann, D., Hufnagel, L., Geisel, T.: The scaling laws of human travel. Nature 439(7075), 462–465 (2006)

    Article  CAS  Google Scholar 

  10. Caballero, M.E., Lambert, A., Bravo, G.U.: Proof(s) of the Lamperti representation of continuous-state branching processes. Probability Surveys 6, 62–89 (2009)

    Article  Google Scholar 

  11. Candia, J., González, M.C., Wang, P., Schoenharl, T., Madey, G., Barabási, A-L.: Uncovering individual and collective human dynamics from mobile phone records. Journal of Physics A, 41(22), 224015 (2008)

    Article  Google Scholar 

  12. Dawson, D.A.: Measure-valued Markov processes. In École d’été de probabilités de Saint-Flour XXI-1991, pp. 1–260. Springer-Verlag, Berlin Heidelberg (1993)

    Chapter  Google Scholar 

  13. Dawson, D.A., Li, Z., Wang, H.: Superprocesses with dependent spatial motion and general branching densities. Electronic Journal of Probability 6, 1–33 (2001)

    Article  Google Scholar 

  14. Engländer, J., Fleischmann, K.: Extinction properties of super-Brownian motions with additional spatially dependent mass production. Stochastic processes and their applications 88(1), 37–58 (2000)

    Article  Google Scholar 

  15. Engländer, J., Kyprianou, A.E.: Local extinction versus local exponential growth for spatial branching processes. The Annals of Probability 32 (1A),78–99 (2004)

    Google Scholar 

  16. Etheridge, A.: An introduction to superprocesses, vol. 20. American Mathematical Society (2000)

    Google Scholar 

  17. Ethier, S.N., Kurtz, T.G.: Markov processes: characterization and convergence, vol. 282. Wiley, New Jersey (2009)

    Google Scholar 

  18. Feller, W.: Diffusion processes in genetics. In Proc. Second Berkeley Symp. Math. Statist. Prob, vol. 227, p. 246 (1951)

    Google Scholar 

  19. Fitzsimmons, P.J.: Construction and regularity of measure-valued Markov branching processes. Israel Journal of Mathematics 64(3), 337–361 (1988)

    Article  Google Scholar 

  20. Fleischmann, K., Klenke, A.: The biodiversity of catalytic super-Brownian motion. Annals of Applied Probability, pp. 1121–1136 (2000)

    Google Scholar 

  21. Gonzalez, M.C., Hidalgo, C.A., Barabasi, A-L.: Understanding individual human mobility patterns. Nature 453(7196), 779–782 (2008)

    Article  CAS  Google Scholar 

  22. Grey, D.R.: Asymptotic behaviour of continuous time, continuous state-space branching processes. Journal of Applied Probability, pp. 669–677 (1974)

    Google Scholar 

  23. Haccou, P., Jagers, P., Vatutin, V.A.: Branching processes: variation, growth, and extinction of populations, vol. 5. Cambridge University Press, Cambridge (2005)

    Book  Google Scholar 

  24. Jiřrina, M.: Stochastic branching processes with continuous state space. Czechoslovak Mathematical Journal 8(2), 292–313 (1958)

    Google Scholar 

  25. Kale, M., Deshmukh, S.R.: Estimation in a continuous state space branching process. Biometrical journal 34(8), 1001–1006 (1992)

    Article  Google Scholar 

  26. Kallenberg, O.: Random measures. Akademie-Verlag, Berlin (1976)

    Google Scholar 

  27. Kashikar, A.S., Deshmukh, S.R.: Second order branching process with continuous state space. Statistics & Probability Letters (2012)

    Google Scholar 

  28. Klein, D.J., Hespanha, J., Madhow, U.: A reaction-diffusion model for epidemic routing in sparsely connected MANETs. In: INFOCOM, 2010a Proceedings IEEE, pp. 1–9 (2010)

    Google Scholar 

  29. Klein, D.J., Hespanha, J., Madhow, U.: A Reaction-diffusion Model For Epidemic Routing In Sparsely Connected MANETs. In: Proceedings of IEEE INFOCOM 2010b, pp. 884–892 (2010)

    Google Scholar 

  30. Kleinrock, L.: Information Flow in Large Communication Nets. Phd thesis proposal, Massachusetts Institute of Technology (1961)

    Google Scholar 

  31. Kleinrock, L.: History of the Internet and its flexible future. Wireless Communications. IEEE 15(1), 8–18 (2008)

    Article  Google Scholar 

  32. Kyprianou, A.E.: Introductory lectures on fluctuations of Lévy processes with applications. Springer-Verlag, Berlin Heidelberg (2007)

    Google Scholar 

  33. Kyprianou, A.E., Liu, R-L., Murillo-Salas, A., Ren, Y-X.: Supercritical super-Brownian motion with a general branching mechanism and travelling waves. Annales de l’Institut Henri Poincaré-Probabilités et Statistiques 48(3), 661–687 (2012)

    Article  Google Scholar 

  34. Lamperti, J.: Continuous-state branching processes. Bull. Amer. Math. Soc, 73(3), 382–386 (1967)

    Article  Google Scholar 

  35. Lamperti, J.: The limit of a sequence of branching processes. Probability Theory and Related Fields 7(4), 271–288 (1967)

    Google Scholar 

  36. Le Gall, J-F.: Spatial branching processes, random snakes and partial differential equations. Birkhäuser-Verlag, Basel (1999)

    Book  Google Scholar 

  37. Le Gall, J-F., Perkins, E.: The Hausdorff measure of the support of two-dimensional super-Brownian motion. The Annals of Probability, pp. 1719–1747 (1995)

    Google Scholar 

  38. Li, Y.: A weak limit theorem for generalized Jiřina processes. Journal of Applied Probability, pp. 453–462 (2009)

    Google Scholar 

  39. Li, Z.: Measure-valued branching Markov processes. Springer-Verlag, Berlin Heidelberg (2011)

    Book  Google Scholar 

  40. Li, Z.: Path-valued branching processes and nonlocal branching superprocesses. arXiv preprint arXiv:1203.6150 (2012)

    Google Scholar 

  41. Li, Z.: Continuous-state branching processes. arXiv:1202.3223 [math.PR] (2012)

    Google Scholar 

  42. Li, Z-H.: Branching processes with immigration and related topics. Frontiers of Mathematics in China 1(1), 73–97 (2006)

    Article  Google Scholar 

  43. Mörters, P.: The average density of super-Brownian motion. Annales de l’Institut Henri Poincare (B) Probability and Statistics 37(1), 71–100 (2001)

    Article  Google Scholar 

  44. Pardoux, E., Wakolbinger, A.: From Brownian motion with a local time drift to Feller’s branching diffusion with logistic growth. Electronic Communications in Probability 16, 720–731 (2011)

    Article  Google Scholar 

  45. Perkins, E.: Dawson-watanabe superprocesses and measure-valued diffusions. Lectures on probability theory and statistics, pp. 125–329 (2002)

    Google Scholar 

  46. Sato, K.: Lévy processes and infinitely divisible distributions. Cambridge University Press, Cambridge, UK (1999)

    Google Scholar 

  47. Sharpe, M.: General theory of Markov processes, vol. 133. Academic Press, Boston, MA (1988)

    Google Scholar 

  48. Silverstein, M.L.: Continuous state branching semigroups. Zeitschrift für Wahrscheinlichkeitstheorie und Verwandte Gebiete 14(2), 96–112 (1969)

    Article  Google Scholar 

  49. Srikant, R.: The mathematics of Internet congestion control. Birkhäuser-Verlag, Boston (2004)

    Book  Google Scholar 

  50. Wang, H.: State classification for a class of interacting superprocesses with location dependent branching. Elect. Commun. Probab 7(16), 157–167 (2002)

    Google Scholar 

  51. Watanabe, S.: On two dimensional Markov processes with branching property. Transactions of the American Mathematical Society 136, 447–466 (1969)

    Article  Google Scholar 

  52. Watson, H.W., Galton, F.: On the probability of the extinction of families. The Journal of the Anthropological Institute of Great Britain and Ireland 4, 138–144 (1875)

    Article  Google Scholar 

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Author information

Authors and Affiliations

  1. Dipartimento di Matematica, Università degli Studi di Torino, via Carlo Alberto 10, I-10123, Torino, Italy

    Laura Sacerdote & Federico Polito

  2. Dipartimento di Informatica, Università degli Studi di Torino, Corso Svizzera 185, I-10149, Torino, Italy

    Michele Garetto & Matteo Sereno

Authors
  1. Laura Sacerdote
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  2. Michele Garetto
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  3. Federico Polito
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  4. Matteo Sereno
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Corresponding author

Correspondence to Laura Sacerdote .

Editor information

Editors and Affiliations

  1. Dipartimento di Matematica, Università degli Studi di Roma Tor Vergata, Roma, Italy

    Alessandra Celletti , Ugo Locatelli  & Elisabetta Strickland ,  & 

  2. Dipartimento di Matematica, Università degli Studi di Bologna, Bologna, Italy

    Tommaso Ruggeri

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Sacerdote, L., Garetto, M., Polito, F., Sereno, M. (2014). Superprocesses as Models for Information Dissemination in the Future Internet. In: Celletti, A., Locatelli, U., Ruggeri, T., Strickland, E. (eds) Mathematical Models and Methods for Planet Earth. Springer INdAM Series, vol 6. Springer, Cham. https://doi.org/10.1007/978-3-319-02657-2_13

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