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Coexistence in Preferential Attachment Networks

Published online by Cambridge University Press:  09 February 2016

University of California, Los Angeles, CA 90025, USA (e-mail:
The Wharton School of the University of Pennsylvania, Philadelphia, PA 19104, USA and University of California, Berkeley, CA 94720, USA (e-mail:
University of California, Berkeley, CA 94720, USA (e-mail:


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 © Cambridge University Press 2016 

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