Abstract
We introduce the perturbed version of the Barabási–Albert random graph with multiple type edges and prove the existence of the (generalized) asymptotic degree distribution. Similarly to the non-perturbed case, the asymptotic degree distribution depends on the almost sure limit of the proportion of edges of different types. However, if there is perturbation, then the resulting degree distribution will be deterministic, which is a major difference compared to the non-perturbed case.
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References
M. A. Abdullah, M. Bode and N. Fountoulakis, Local majority dynamics on preferential attachment graphs, in: International Workshop on Algorithms and Models for the Web-Graph, Springer (Cham, 2015)
Antunović, T., Mossel, E., Rácz, M.: Coexistence in preferential attachment networks. Combin. Probab. Comput. 25, 797–822 (2016)
Á. Backhausz and B. Rozner, Asymptotic degree distribution in preferential attachment graph models with multiple type edges, Stoch. Models (2019), 1–27.
Barabási, A.-L., Albert, R.: Emergence of scaling in random networks. Science 286, 509–512 (1999)
B. Bollobás, C. Borgs, J. Chayes and O. Riordan, Directed scale-free graphs, in: Proceedings of the 14th Annual ACM-SIAM Symposium on Discrete Algorithms (SODA) (2003), pp. 132-139
Bollobás, B., Riordan, O., Spencer, J., Tusnády, G.: The degree sequence of a scale-free random graph process. Random Structures Algorithms 18, 279–290 (2001)
Durrett, R.: Random Graph Dynamics, Cambridge Series in Statistical and Probabilistic Mathematics. Cambridge Univ, Press (Cambridge (2007)
A. Frieze and M. Karoński, Introduction to Random Graphs, Cambridge University Press (Cambridge, 2015)
U. Gangopadhyay and K. Maulik, Stochastic approximation with random step sizes and urn models with random replacement matrices, arXiv:1709.00467
van der Hofstad, R.: Random Graphs and Complex Networks, Cambridge Series in Statistical and Probabilistic Mathematics, vol. 43. Cambridge Univ, Press (Cambridge (2016)
Laruelle, S., Pagès, G.: Randomized urn models revisited using stochastic approximation. Ann. Appl. Probab. 23, 1409–1436 (2013)
L. Ostroumova, A. Ryabchenko and E. Samosvat, Generalized preferential attachment: tunable powerlaw degree distribution and clustering coefficient, in: Algorithms and Models for the Web Graph, Lecture Notes in Comput. Sci., Springer (Cham), pp. 185–202
S. Rosengren, A multi-type preferential attachment model, Internet Mathematics (2018).
Wang, T., Resnick, S.: Degree growth rates and index estimation in a directed preferential attachment model. Stochastic Process, Appl (2019)
Acknowledgements
This research was partially supported by Pallas Athene Domus Educationis Foundation. The views expressed are those of the authors’ and do not necessarily reflect the official opinion of Pallas Athene Domus Educationis Foundation. The first author was supported by the Bolyai Research Grant of the Hungarian Academy of Sciences.
The authors thank the referee for the review and appreciate the comments and suggestions which contributed to the improvements of the article.
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Backhausz, Á., Rozner, B. Barabási–Albert random graph with multiple type edges and perturbation. Acta Math. Hungar. 161, 212–229 (2020). https://doi.org/10.1007/s10474-019-01005-5
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DOI: https://doi.org/10.1007/s10474-019-01005-5