Advances in Applied Probability

Distance between two random k-out digraphs, with and without preferential attachment

Nicholas R. Peterson and Boris Pittel

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A random k-out mapping (digraph) on [n] is generated by choosing k random images of each vertex one at a time, subject to a 'preferential attachment' rule: the current vertex selects an image i with probability proportional to a given parameter α = α(n) plus the number of times i has already been selected. Intuitively, the larger α becomes, the closer the resulting k-out mapping is to the uniformly random k-out mapping. We prove that α = Θ(n1/2) is the threshold for α growing 'fast enough' to make the random digraph approach the uniformly random digraph in terms of the total variation distance. We also determine an exact limit for this distance for the α = β n1/2 case.

Article information

Adv. in Appl. Probab., Volume 47, Number 3 (2015), 858-879.

First available in Project Euclid: 8 October 2015

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 05C80: Random graphs [See also 60B20]
Secondary: 60C05: Combinatorial probability 60F05: Central limit and other weak theorems

Random graphs random digraphs preferential attachment uniform k-out digraphs total variation distance local limit theorem


Peterson, Nicholas R.; Pittel, Boris. Distance between two random k -out digraphs, with and without preferential attachment. Adv. in Appl. Probab. 47 (2015), no. 3, 858--879. doi:10.1239/aap/1444308885.

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  • Barabási, A.-L. and Albert, R. (1999). Emergence of scaling in random networks. Science 286, 509–512.
  • Bergeron, F., Flajolet, P. and Salvy, B. (1992). Varieties of increasing trees. In CAAP '92 (Lecture Notes Comput. Sci. 581), Springer, Berlin, pp. 24–48.
  • Bhattacharya, R. N. and Rao, R. R. (1976). Normal Approximation and Asymptotic Expansions. John Wiley, New York.
  • Bollobás, B. and Riordan, O. (2004). The diameter of a scale-free random graph. Combinatorica 24, 5–34.
  • Bollobás, B., Borgs, C., Chayes, J. and Riordan, O. (2003). Directed scale-free graphs. In Proc. 14th ACM-SIAM Symposium on Discrete Algorithms, ACM, New York, pp. 132–139.
  • Bollobás, B., Riordan, O., Spencer, J. and Tusnády, G. (2001). The degree sequence of a scale-free random graph process. Random Structures Algorithms 18, 279–290.
  • Buckley, P. G. and Osthus, D. (2004). Popularity based random graph models leading to a scale-free degree sequence. Discrete Math. 282, 53–68.
  • Burtin, Y. D. (1980). On a simple formula for random mappings and its applications. J. Appl. Prob. 17, 403–414.
  • Deijfen, M. (2010). Random networks with preferential growth and vertex death. J. Appl. Prob. 47, 1150–1163.
  • Durrett, R. (2005). Probability: Theory and Examples, 3rd edn. Thomson Brooks/Cole, Belmont, CA.
  • Erdős, P. and Rényi, A. (1960). On the evolution of random graphs. Magyar. Tud. Akad. Kutató Int. Közl. 5, 17–61.
  • Gertsbakh, I. B. (1977). Epidemic process on a random graph: some preliminary results. J. Appl. Prob. 14, 427–438.
  • Hansen, J. C. and Jaworski, J. (2008). Local properties of random mappings with exchangeable in-degrees. Adv. Appl. Prob. 40, 183–205.
  • Hansen, J. C. and Jaworski, J. (2008). Random mappings with exchangeable in-degrees. Random Structures Algorithms 33, 105–126.
  • Hansen, J. C. and Jaworski, J. (2009). A random mapping with preferential attachment. Random Structures Algorithms 34, 87–111.
  • Mahmoud, H. M., Smythe, R. T. and Szymański, J. (1993). On the structure of random plane-oriented recursive trees and their branches. Random Structures Algorithms 4, 151–176.
  • Pittel, B. (1983). On distributions related to transitive closures of random finite mappings. Ann. Prob. 11, 428–441.
  • Pittel, B. (1994). Note on the heights of random recursive trees and random $m$-ary search trees. Random Structures Algorithms 5, 337–347.
  • Pittel, B. (2010). On a random graph evolving by degrees. Adv. Math. 223, 619–671.