publication . Preprint . Other literature type . Article . 2016

Generalized Nonlinear Yule Models

Petr Lansky; Federico Polito; Laura Sacerdote;
Open Access English
  • Published: 04 Oct 2016
Abstract
With the aim of considering models related to random graphs growth exhibiting persistent memory, we propose a fractional nonlinear modification of the classical Yule model often studied in the context of macroevolution. Here the model is analyzed and interpreted in the framework of the development of networks such as the World Wide Web. Nonlinearity is introduced by replacing the linear birth process governing the growth of the in-links of each specific webpage with a fractional nonlinear birth process with completely general birth rates. Among the main results we derive the explicit distribution of the number of in-links of a webpage chosen uniformly at random ...
Subjects
free text keywords: Mathematics - Probability, 60G22, 60J80, 05C80, Mathematical Physics, Statistical and Nonlinear Physics, Random graph, Mean value, Mathematics, Fractional calculus, Calculus, Asymptotic analysis, Nonlinear system, Macroevolution, Applied mathematics, Point process, Fractional Poisson process
Related Organizations
40 references, page 1 of 3

[1] N. T. J. Bailey. The elements of stochastic processes with applications to the natural sciences. John Wiley & Sons, Inc., New York-London-Sydney, 1964.

[2] A.-L. Baraba´si and R. Albert. Emergence of scaling in random networks. Science, 286(5439): 509-512, 1999. doi: 10.1126/science.286.5439.509.

[3] L. Beghin and M. D'Ovidio. Fractional Poisson process with random drift. Electron. J. Probab., 19:no. 122, 26, 2014. doi: 10.1214/EJP.v19-3258. [OpenAIRE]

[4] L. Beghin and E. Orsingher. Fractional Poisson processes and related planar random motions. Electron. J. Probab., 14:no. 61, 1790-1827, 2009. doi: 10.1214/EJP.v14-675. [OpenAIRE]

[5] J. Bertoin. L´evy processes, volume 121 of Cambridge Tracts in Mathematics. Cambridge University Press, Cambridge, 1996. ISBN 0-521-56243-0.

[6] D. O. Cahoy and F. Polito. On a fractional binomial process. J. Stat. Phys., 146(3):646-662, 2012. doi: 10.1007/s10955-011-0408-3.

[7] D. O. Cahoy and F. Polito. Simulation and estimation for the fractional Yule process. Methodol. Comput. Appl. Probab., 14(2):383-403, 2012. doi: 10.1007/s11009-010-9207-6.

[8] D. O. Cahoy and F. Polito. Renewal processes based on generalized Mittag-Leffler waiting times. Commun. Nonlinear Sci. Numer. Simul., 18(3):639-650, 2013. doi: 10.1016/j.cnsns.2012.08.013. [OpenAIRE]

[9] D. O. Cahoy and F. Polito. Parameter estimation for fractional birth and fractional death processes. Stat. Comput., 24(2):211-222, 2014. doi: 10.1007/s11222-012-9365-1.

[10] K. S. Crump. On point processes having an order statistic structure. Sankhya¯ Ser. A, 37(3): 396-404, 1975.

[11] K. S. Crump. On point processes having an order statistic structure. Sankhya¯ Ser. A, 37(3): 396-404, 1975.

[12] K. Diethelm. The analysis of fractional differential equations, volume 2004 of Lecture Notes in Mathematics. Springer-Verlag, Berlin, 2010. ISBN 978-3-642-14573-5. doi: 10.1007/978- 3-642-14574-2. An application-oriented exposition using differential operators of Caputo type.

[13] P. D. Feigin. On the characterization of point processes with the order statistic property. J. Appl. Probab., 16(2):297-304, 1979.

[14] P. D. Feigin and B. Reiser. On asymptotic ancillarity and inference for Yule and regular nonergodic processes. Biometrika, 66(2):279-283, 1979. doi: 10.1093/biomet/66.2.279.

[15] R. Garra, R. Gorenflo, F. Polito, and Zˇ. Tomovski. Hilfer-Prabhakar derivatives and some applications. Appl. Math. Comput., 242:576-589, 2014. doi: 10.1016/j.amc.2014.05.129. [OpenAIRE]

40 references, page 1 of 3
Any information missing or wrong?Report an Issue