Random walk attachment graphs

Chris Cannings (University of Sheffield)
Jonathan H Jordan (University of Sheffield)


We consider the random walk attachment graph introduced by Saramäki and Kaski and proposed as a mechanism to explain how behaviour similar to preferential attachment may appear requiring only local knowledge.  We show that if the length of the random walk is fixed then the resulting graphs can have properties significantly different from those of preferential attachment graphs, and in particular that in the case where the random walks are of length 1 and each new vertex attaches to a single existing vertex the proportion of vertices which have degree 1 tends to 1, in contrast to preferential attachment models.

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Pages: 1-5

Publication Date: September 18, 2013

DOI: 10.1214/ECP.v18-2518


  • R. Albert, A.-L. Barabási, and H. Jeong. Mean-field theory for scale-free random networks. Physica A, 272:173--187, 1999.
  • Barabási, Albert-László; Albert, Réka. Emergence of scaling in random networks. Science 286 (1999), no. 5439, 509--512. MR2091634
  • Bollobás, Béla; Riordan, Oliver; Spencer, Joel; Tusnády, Gábor. The degree sequence of a scale-free random graph process. Random Structures Algorithms 18 (2001), no. 3, 279--290. MR1824277
  • Dereich, Steffen; Mörters, Peter. Random networks with sublinear preferential attachment: degree evolutions. Electron. J. Probab. 14 (2009), no. 43, 1222--1267. MR2511283
  • Dereich, Steffen; Mörters, Peter. Random networks with concave preferential attachment rule. Jahresber. Dtsch. Math.-Ver. 113 (2011), no. 1, 21--40. MR2760002
  • Evans, T. S.; Saramäki, J. P. Scale-free networks from self-organization. Phys. Rev. E (3) 72 (2005), no. 2, 026138, 14 pp. MR2177389
  • Freedman, David A. Bernard Friedman's urn. Ann. Math. Statist 36 1965 956--970. MR0177432
  • Pemantle, Robin. A survey of random processes with reinforcement. Probab. Surv. 4 (2007), 1--79. MR2282181
  • Saramäki, Jari; Kaski, Kimmo. Scale-free networks generated by random walkers. Phys. A 341 (2004), no. 1-4, 80--86. MR2092677
  • Simon, Herbert A. On a class of skew distribution functions. Biometrika 42 (1955), 425--440. MR0073085
  • G. U. Yule. A mathematical theory of evolution, based on the conclusions of Dr. J. C. Willis, F.R.S. Philosophical Transactions of the Royal Society of London, B, 213:21--87, 1925.

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