Skip to main content Accessibility help

Information Transmission under Random Emission Constraints



We model the transmission of a message on the complete graph with n vertices and limited resources. The vertices of the graph represent servers that may broadcast the message at random. Each server has a random emission capital that decreases at each emission. Quantities of interest are the number of servers that receive the information before the capital of all the informed servers is exhausted and the exhaustion time. We establish limit theorems (law of large numbers, central limit theorem and large deviation principle), as n → ∞, for the proportion of informed vertices before exhaustion and for the total duration. The analysis relies on a construction of the transmission procedure as a dynamical selection of successful nodes in a Galton–Watson tree with respect to the success epochs of the coupon collector problem.



Hide All
[1]Alves, O., Lebensztayn, E., Machado, F. and Martinez, M. (2006) Random walks systems on complete graphs. Bull. Braz. Math. Soc. (NS) 37 571580.
[2]Alves, O., Machado, F. and Popov, S. (2002) Phase transition for the frog model. Electron. J. Probab. 7 #16.
[3]Alves, O., Machado, F. and Popov, S. (2002) The shape theorem for the frog model. Ann. Appl. Probab. 12 533546.
[4]Baccelli, F., Blaszczyszyn, B. and Mirsadeghi, M. (2011) Optimal paths on the space–time SINR random graph, Adv. Appl. Probab. 43 131150.
[5]Baum, L. and Billingsley, P. (1965) Asymptotic distributions for the coupon collector's problem. Ann. Math. Statist. 36 18351839.
[6]Billingsley, P. (1968) Convergence of Probability Measures, Wiley.
[7]Boucheron, S., Gamboa, F. and Léonard, C. (2002) Bins and balls: Large deviations of the empirical occupancy process. Ann. Appl. Probab. 12 607636.
[8]Comets, F., Quastel, J. and Ramírez, A. (2007) Fluctuations of the front in a stochastic combustion model. Ann. Inst. H. Poincaré Probab. Statist. 43 147162.
[9]Comets, F., Quastel, J. and Ramírez, A. (2009) Fluctuations of the front in a one dimensional model of X+Y → 2X. Trans. Amer. Math. Soc. 361 61656189.
[10]Dacunha-Castelle, D. and Duflo, M. (1983) Probabilités et Statistiques 2: Temps Mobile, Masson.
[11]Dembo, A. and Zeitouni, O. (1998) Large Deviations Techniques and Applications, second edition, Springer.
[12]Ding, L. and Guan, Z.-H. (2008) Modeling wireless sensor networks using random graph theory. Physica A 387 30083016.
[13]Dupuis, P., Nuzman, C. and Whiting, P. (2004) Large deviation asymptotics for occupancy problems. Ann. Probab. 32 27652818.
[14]Duquesne, T. and Le Gall, J.-F. (2002) Random trees, Lévy processes and spatial branching processes. Astérisque 281.
[15]Durrett, R. (1995) Probability: Theory and Examples, second edition, Duxbury.
[16]Erdös, P. and Rényi, A. (1961) On a classical problem of probability theory. Magyar Tud. Akad. Mat. Kutató Int. Közl. 6 215220.
[17]Flajolet, P., Gardy, D. and Thimonier, L. (1992) Birthday paradox, coupon collectors, caching algorithms and self-organizing search. Discrete Appl. Math. 39 207229.
[18]Jacod, J. and Shiryaev, A. N. (2002) Limit Theorems for Stochastic Processes, second edition, Springer.
[19]Jia, X. (2004) Wireless networks and random geometric graphs. In Proc. 2004 International Symposium on Parallel Architectures, Algorithms and Networks: ISPAN'04, pp. 575–580.
[20]Kan, N. (2002) The martingale approach to the coupon collection problem. Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. 294 113126. Translation in J. Math. Sci. 127 (2005) 1737–1744.
[21]Kawahigashi, H., Terashima, Y., Miyauchi, N. and Nakakawaji, T. (2005) Modeling ad hoc sensor networks using random graph theory. In Proc. Second IEEE Consumer Communications and Networking Conference: CCNC 2005.
[22]Kesten, H. and Sidoravicius, V. (2005) The spread of a rumor or infection in a moving population. Ann. Probab. 33 24022462.
[23]Kesten, H. and Sidoravicius, V. (2006) A phase transition in a model for the spread of an infection. Illinois J. Math. 50 547634.
[24]Kesten, H. and Stigum, B. P. (1966) A limit theorem for multidimensional Galton–Watson processes. Ann. Math. Statist. 37 12111223.
[25]Kurkova, I., Popov, S. and Vachkovskaia, M. (2004) On infection spreading and competition between independent random walks. Electron. J. Probab. 9 122.
[26]Kurtz, T., Lebensztayn, E., Leichsenring, A. R. and Machado, F. P. (2008) Limit theorems for an epidemic model on the complete graph. ALEA Lat. Am. J. Probab. Math. Stat. 4 4555.
[27]Machado, F., Machurian, H. and Matzinger, H. (2011) CLT for the proportion of infected individuals for an epidemic model on a complete graph. Markov Proc. Rel. Fields 17 209224.
[28]Neveu, J. (1986) Arbres et processus de Galton–Watson. Annales de l'IHP B 22 199207.
[29]Pitman, J. (2006) Combinatorial Stochastic Processes: St. Flour 2002, Vol. 1875 of Lecture Notes in Mathematics, Springer.
[30]Ramírez, A. and Sidoravicius, V. (2004) Asymptotic behavior of a stochastic combustion growth process. J. Eur. Math. Soc. 6 293334.
[31]Sedgewick, R. and Flajolet, P. (1996) An Introduction to the Analysis of Algorithms, Addison-Wesley.
[32]Zhukovskiï, M. E. (2012) The law of large numbers for an epidemic model (Russian). Dokl. Akad. Nauk 442 736739. Translation in Dokl. Math. 85 (2012) 113–116.


Information Transmission under Random Emission Constraints



Altmetric attention score

Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed