Skip to main content
×
×
Home

Coexistence in Preferential Attachment Networks

  • TONĆI ANTUNOVIĆ (a1), ELCHANAN MOSSEL (a2) and MIKLÓS Z. RÁCZ (a3)
Abstract

We introduce a new model of competition on growing networks. This extends the preferential attachment model, with the key property that node choices evolve simultaneously with the network. When a new node joins the network, it chooses neighbours by preferential attachment, and selects its type based on the number of initial neighbours of each type. The model is analysed in detail, and in particular, we determine the possible proportions of the various types in the limit of large networks. An important qualitative feature we find is that, in contrast to many current theoretical models, often several competitors will coexist. This matches empirical observations in many real-world networks.

Copyright
References
Hide All
[1] Antunović, T., Dekel, Y., Mossel, E. and Peres, Y. (2011) Competing first passage percolation on random regular graphs. arXiv:1109.2575
[2] Arthur, W. B. (1990) Positive feedbacks in the economy. Sci. Am. 262 9299.
[3] Arthur, W. B. (1994) Increasing Returns and Path Dependence in the Economy, The University of Michigan Press.
[4] Banerjee, A. and Fudenberg, D. (2004) Word-of-mouth learning. Game. Econ. Behav. 46 122.
[5] Barabási, A. L. and Albert, R. (1999) Emergence of scaling in random networks. Science 286 (5439) 509512.
[6] Barrat, A., Barthélemy, M. and Vespignani, A. (2008) Dynamical Processes on Complex Networks, Cambridge University Press.
[7] Benaïm, M. (1999) Dynamics of stochastic approximation algorithms. In Séminaire de Probabilités XXXIII, Vol. 1709 of Lecture Notes in Mathematics, Springer, pp. 168.
[8] Benaïm, M., Benjamini, I., Chen, J. and Lima, Y. (2015) A generalized Pólya's urn with graph based interactions. Random Struct. Alg. 46 614634.
[9] Benaïm, M. and Hirsch, M. W. (1996) Asymptotic pseudotrajectories and chain recurrent flows, with applications. J. Dynam. Differential Equations 8 141176.
[10] Berger, N., Borgs, C., Chayes, J. T. and Saberi, A. (2014) Asymptotic behavior and distributional limits of preferential attachment graphs. Ann. Probab. 42 140.
[11] Bollobás, B. and Riordan, O. (2004) The diameter of a scale-free random graph. Combinatorica 24 534.
[12] Deijfen, M. and van der Hofstad, R. (2013) The winner takes it all. arXiv:1306.6467.
[13] Gross, T. and Blasius, B. (2008) Adaptive coevolutionary networks: A review. J. Roy. Soc. Interface 5 259271.
[14] Hill, B. M., Lane, D. and Sudderth, W. (1980) A strong law for some generalized urn processes. Ann. Probab. 8 214226.
[15] Hirsch, M. W., Smale, S. and Devaney, R. L. (2004) Differential Equations, Dynamical Systems, and An Introduction to Chaos, Academic Press.
[16] Holme, P. and Saramäki, J. (2012) Temporal networks. Phys. Rep. 519 97125.
[17] Lelarge, M. (2012) Diffusion and cascading behavior in random networks. Games and Economic Behavior 75 752775.
[18] Nevelson, M. B. and Hasminskii, R. Z. (1976) Stochastic Approximation and Recursive Estimation , Vol. 47 of Translations of Mathematical Monographs, AMS.
[19] Ohtsuki, H., Hauert, C., Lieberman, E. and Nowak, M. A. (2006) A simple rule for the evolution of cooperation on graphs and social networks. Nature 441 (7092) 502505.
[20] Pastor-Satorras, R. and Vespignani, A. (2001) Epidemic spreading in scale-free networks. Phys. Rev. Lett. 86 32003203.
[21] Pemantle, R. (1990) A time-dependent version of Pólya's urn. J. Theoret. Probab. 3 627637.
[22] Pemantle, R. (1990) Non-convergence to unstable points in urn models and stochastic approximations. Ann. Probab. 18 698712.
[23] Pemantle, R. (1991) When are touchpoints limits for generalized Pólya urns? Proc. Amer. Math. Soc. 113 235243.
[24] Pemantle, R. (2007) A survey of random processes with reinforcement. Probab. Surv. 4 179.
[25] Prakash, B. A., Beutel, A., Rosenfeld, R. and Faloutsos, C. (2012) Winner takes all: competing viruses or ideas on fair-play networks. In Proc. 21st Int. Conf. World Wide Web (WWW), ACM, pp. 10371046.
[26] Redner, S. (1998) How popular is your paper? An empirical study of the citation distribution. Eur. Phys. J. B 4 131134.
[27] Robbins, H. and Monro, S. (1951) A stochastic approximation method. Ann. Math. Statist. 22 400407.
[28] Shaked, M. and Shanthikumar, J. G. (2007) Stochastic Orders, Springer.
[29] Skyrms, B. and Pemantle, R. (2000) A dynamic model of social network formation. Proc. Nat. Acad. Sci. USA 97 93409346.
[30] Watts, D. J. (2002) A simple model of global cascades on random networks. Proc. Nat. Acad. Sci. USA 99 57665771.
Recommend this journal

Email your librarian or administrator to recommend adding this journal to your organisation's collection.

Combinatorics, Probability and Computing
  • ISSN: 0963-5483
  • EISSN: 1469-2163
  • URL: /core/journals/combinatorics-probability-and-computing
Please enter your name
Please enter a valid email address
Who would you like to send this to? *
×

MSC classification

Metrics

Full text views

Total number of HTML views: 0
Total number of PDF views: 76 *
Loading metrics...

Abstract views

Total abstract views: 346 *
Loading metrics...

* Views captured on Cambridge Core between September 2016 - 23rd September 2018. This data will be updated every 24 hours.