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 tends to infinity, for the proportion of visited 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

Rumor Processes On ℕ And Discrete Renewal Processes

We study two rumor processes on ℕ, the dynamics of which are related to an SI epidemic model with long range transmission. Both models start with one spreader at site 0 and ignorants at all the other sites of ℕ, but differ by the transmission mechanism. In one model, the spreaders transmit the information within a random distance on their right, and in the other the ignorants take the information from a spreader within a random distance on their left. We obtain the probability of survival, information on the distribution of the range of the rumor and limit theorems for the proportion of spreaders. The key step of our proofs is to show that, in each model, the position of the spreaders on ℕ can be related to a suitably chosen discrete renewal process. © 2014 Springer Science+Business Media New York.1553591602Andersson, H., Limit theorems for a random graph epidemic model (1998) Ann. Appl. Probab., 8 (4), pp. 1331-1349Athreya, S., Roy, R., Sarkar, A., On the coverage of space by random sets (2004) Adv. Appl. Probab., 36 (1), pp. 1-18Berger, N., Borgs, C., Chayes, J.T., Saberi, A., On the spread of viruses on the internet (2005) Proceedings of the Sixteenth Annual ACM-SIAM Symposium on Discrete Algorithms, , SODA '05Bertacchi, D., Zucca, F., Rumor processes in random environment on ℕ and galton-watson trees (2013) J. Stat. Phys., 153 (3), pp. 486-511Bressaud, X., Fernández, R., Galves, A., Decay of correlations for non-Hölderian dynamics. A coupling approach (1999) Elect. J. Probab, 4 (3), pp. 1-19Coletti, C.F., Rodríguez, P.M., Schinazi, R.B., A spatial stochastic model for rumor transmission (2012) J. Stat. Phys., 147 (2), pp. 375-381Comets, F., Delarue, D., Schott, R., Information transmission under random emission constraints Comb. Probab. Comput, , arXiv: 1309. 0624 (2013, to appear)Comets, F., Fernandez, R., Ferrari, P., Processes with long memory: regenerative construction and perfect simulation (2002) Ann. Appl. Prob., 12 (3), pp. 921-943Daley, D.J., Kendall, D.G., Stochastic rumours (1965) J. Inst. Math. Appl., 1, pp. 42-55Durrett, R., Jung, P., Two phase transitions for the contact process on small worlds (2007) Stoch. Process. Appl., 117 (12), pp. 1910-1927Gallo, S., Lerasle, M., Takahashi, Y.D., Markov approximation of chains of infinite order in the d-metric (2013) Markov Process. Relat. Fields, 19 (1), pp. 51-82Garsia, A., Lamperti, J., A discrete renewal theorem with infinite mean (1962) Comment. Math. Helv., 37, pp. 221-234Isham, V., Harden, S., Nekovee, M., Stochastic epidemics and rumours on finite random networks (2010) Phys. A: Stat. Mech. Appl., 389 (3), pp. 561-576Junior, V.V., Machado, F.P., Zuluaga, M., Rumor processes on ℕ (2011) J. Appl. Probab., 48 (3), pp. 624-636Kurtz, T.G., Lebensztayn, E., Leichsenring, A.R., Machado, F.P., Limit theorems for an epidemic model on the complete graph. ALEA Lat (2008) Am. J. Probab. Math. Stat., 4, pp. 45-55Lebensztayn, E., Machado, F.P., Rodríguez, P.M., Limit theorems for a general stochastic rumour model (2011) SIAM J. Appl. Math, 71 (4), pp. 1476-1486Lebensztayn, E., Machado, F.P., Rodríguez, P.M., Process with random stifling (2011) Environ. Modell. Softw, 26 (4), pp. 517-522Maki, D.P., Thompson, M., (1973) Mathematical Models and Applications: With Emphasis on the Social, Life, and Management Sciences, , Englewood Cliffs: Prentice-Hall IncMoreno, Y., Nekovee, M., Pacheco, A.F., Dynamics of rumor spreading in complex networks (2004) Phys. Rev. E, 69, p. 066130Pittel, B., On spreading a rumor (1987) SIAM J. Appl. Math., 47 (1), pp. 213-223Ross, S.M., (2009) Introduction to Probability Models, , 10th edn., Burlington: Academic Press In