The Annals of Applied Probability

Twitter event networks and the Superstar model

Shankar Bhamidi, J. Michael Steele, and Tauhid Zaman

Full-text: Open access

Abstract

Condensation phenomenon is often observed in social networks such as Twitter where one “superstar” vertex gains a positive fraction of the edges, while the remaining empirical degree distribution still exhibits a power law tail. We formulate a mathematically tractable model for this phenomenon that provides a better fit to empirical data than the standard preferential attachment model across an array of networks observed in Twitter. Using embeddings in an equivalent continuous time version of the process, and adapting techniques from the stable age-distribution theory of branching processes, we prove limit results for the proportion of edges that condense around the superstar, the degree distribution of the remaining vertices, maximal nonsuperstar degree asymptotics and height of these random trees in the large network limit.

Article information

Source
Ann. Appl. Probab., Volume 25, Number 5 (2015), 2462-2502.

Dates
Received: November 2012
Revised: July 2014
First available in Project Euclid: 30 July 2015

Permanent link to this document
https://projecteuclid.org/euclid.aoap/1438261046

Digital Object Identifier
doi:10.1214/14-AAP1053

Mathematical Reviews number (MathSciNet)
MR3375881

Zentralblatt MATH identifier
1334.60204

Subjects
Primary: 60C05: Combinatorial probability 05C80: Random graphs [See also 60B20] 90B15: Network models, stochastic

Keywords
Dynamic networks preferential attachment continuous time branching processes characteristics of branching processes multitype branching processes Twitter social networks retweet graph

Citation

Bhamidi, Shankar; Steele, J. Michael; Zaman, Tauhid. Twitter event networks and the Superstar model. Ann. Appl. Probab. 25 (2015), no. 5, 2462--2502. doi:10.1214/14-AAP1053. https://projecteuclid.org/euclid.aoap/1438261046


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References

  • [1] Asur, S., Huberman, B. A., Szabo, G. and Wang, C. (2011). Trends in social media: Persistence and decay. In AAAI Conference on Weblogs and Social Media, Barcelona, Spain.
  • [2] Athreya, K. B., Ghosh, A. P. and Sethuraman, S. (2008). Growth of preferential attachment random graphs via continuous-time branching processes. Proc. Indian Acad. Sci. Math. Sci. 118 473–494.
  • [3] Athreya, K. B. and Ney, P. E. (1972). Branching Processes. Springer, New York.
  • [4] Barabási, A.-L. and Albert, R. (1999). Emergence of scaling in random networks. Science 286 509–512.
  • [5] 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.
  • [6] Bollobás, B. and Riordan, O. M. (2003). Mathematical results on scale-free random graphs. In Handbook of Graphs and Networks 1–34. Wiley, Weinheim.
  • [7] Cha, M., Haddadi, H., Benevenuto, F. and Gummadi, K. P. (2010). Measuring user influence in Twitter: The million follower fallacy. In AAAI Conference on Weblogs and Social Media, Washington, DC.
  • [8] Cooper, C. and Frieze, A. (2003). A general model of web graphs. Random Structures Algorithms 22 311–335.
  • [9] Corless, R. M., Gonnet, G. H., Hare, D. E. G., Jeffrey, D. J. and Knuth, D. E. (1996). On the Lambert $W$ function. Adv. Comput. Math. 5 329–359.
  • [10] Data courtesy Microsoft Research Cambridge. August 2010.
  • [11] Deijfen, M. (2010). Random networks with preferential growth and vertex death. J. Appl. Probab. 47 1150–1163.
  • [12] Dorogovtsev, S. N. and Mendes, J. F. F. (2003). Evolution of Networks. Oxford Univ. Press, Oxford.
  • [13] Durrett, R. (2007). Random Graph Dynamics. Cambridge Univ. Press, Cambridge.
  • [14] Jagers, P. (1975). Branching Processes with Biological Applications. Wiley, London.
  • [15] Jagers, P. and Nerman, O. (1984). The growth and composition of branching populations. Adv. in Appl. Probab. 16 221–259.
  • [16] Jagers, P. and Nerman, O. (1996). The asymptotic composition of supercritical multi-type branching populations. In Séminaire de Probabilités, XXX. Lecture Notes in Math. 1626 40–54. Springer, Berlin.
  • [17] Kingman, J. F. C. (1975). The first birth problem for an age-dependent branching process. Ann. Probab. 3 790–801.
  • [18] Kwak, H., Lee, C., Park, H. and Moon, S. (2010). What is Twitter, a social network or a news media? In Proc. WWW. Proceedings of the 19th International Conference on World Wide Web 591–600. ACM, New York.
  • [19] Móri, T. F. (2007). Degree distribution nearby the origin of a preferential attachment graph. Electron. Commun. Probab. 12 276–282 (electronic).
  • [20] Nerman, O. (1981). On the convergence of supercritical general (C–M–J) branching processes. Z. Wahrsch. Verw. Gebiete 57 365–395.
  • [21] Newman, M. E. J. (2003). The structure and function of complex networks. SIAM Rev. 45 167–256 (electronic).
  • [22] Norris, J. R. (1998). Markov Chains. Cambridge Series in Statistical and Probabilistic Mathematics 2. Cambridge Univ. Press, Cambridge.
  • [23] Pittel, B. (1994). Note on the heights of random recursive trees and random $m$-ary search trees. Random Structures Algorithms 5 337–347.
  • [24] Rudas, A. and Tóth, B. (2009). Random tree growth with branching processes—a survey. In Handbook of Large-Scale Random Networks. Bolyai Soc. Math. Stud. 18 171–202. Springer, Berlin.
  • [25] Rudas, A., Tóth, B. and Valkó, B. (2007). Random trees and general branching processes. Random Structures Algorithms 31 186–202.
  • [26] Smythe, R. T. and Mahmoud, H. M. (1995). A survey of recursive trees. Theory Probab. Math. Statist. 51 1–27.
  • [27] Szymański, J. (1987). On a nonuniform random recursive tree. In Random Graphs’85 (Poznań, 1985). North-Holland Math. Stud. 144 297–306. North-Holland, Amsterdam.
  • [28] Twitter developers. Available at https://dev.twitter.com/docs/api/1.1/get/statuses/firehose, August 2012.
  • [29] Wiuf, C., Brameier, M., Hagberg, O. and Stumpf, M. P. H. (2006). A likelihood approach to analysis of network data. Proc. Natl. Acad. Sci. USA 103 7566–7570 (electronic).