Hostname: page-component-586b7cd67f-dsjbd Total loading time: 0 Render date: 2024-11-24T00:56:40.071Z Has data issue: false hasContentIssue false
  • English
  • Français

Measuring reciprocity in a directed preferential attachment network

Part of: Graph theory Stochastic processes Limit theorems

Published online by Cambridge University Press:  15 June 2022

Tiandong Wang Open the ORCID record for Tiandong Wang [Opens in a new window]  and
Sidney I. Resnick
Tiandong Wang*
Affiliation:
Texas A&M University
Sidney I. Resnick*
Affiliation:
Cornell University
*
*Postal address: Department of Statistics, Texas A&M University, College Station, TX 77843, U.S.A. Email address: [email protected]
**Postal address: School of Operations Research and Information Engineering, Cornell University, Ithaca, NY 14853, U.S.A. Email address: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

Empirical studies (e.g. Jiang et al. (2015) and Mislove et al. (2007)) show that online social networks have not only in- and out-degree distributions with Pareto-like tails, but also a high proportion of reciprocal edges. A classical directed preferential attachment (PA) model generates in- and out-degree distributions with power-law tails, but the theoretical properties of the reciprocity feature in this model have not yet been studied. We derive asymptotic results on the number of reciprocal edges between two fixed nodes, as well as the proportion of reciprocal edges in the entire PA network. We see that with certain choices of parameters, the proportion of reciprocal edges in a directed PA network is close to 0, which differs from the empirical observation. This points out one potential problem of fitting a classical PA model to a given network dataset with high reciprocity, and indicates that alternative models need to be considered.

Keywords

Reciprocitypreferential attachmentin- and out-degrees

MSC classification

Primary: 60F15: Strong theorems 60G46: Martingales and classical analysis
Secondary: 05C82: Small world graphs, complex networks
Type
Original Article
Information
Advances in Applied Probability , Volume 54 , Issue 3 , September 2022 , pp. 718 - 742
Check for updates
Copyright
© The Author(s), 2022. Published by Cambridge University Press on behalf of Applied Probability Trust

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Athreya, K. and Ney, P. (2004). Branching Processes. Springer, New York.Google Scholar
Bollobás, B., Borgs, C., Chayes, J. and Riordan, O. (2003). Directed scale-free graphs. In Proceedings of the Fourteenth Annual ACM–SIAM Symposium on Discrete Algorithms (Baltimore, 2003), Association for Computing Machinery, New York, pp. 132–139.Google Scholar
Cha, M., Mislove, A. and Gummadi, K. (2009). A measurement-driven analysis of information propagation in the Flickr social network. In Proceedings of the 18th International Conference on World Wide Web (WWW ’09), Association for Computing Machinery, New York, pp. 721–730.10.1145/1526709.1526806CrossRefGoogle Scholar
Cheng, J., Romero, D., Meeder, B. and Kleinberg, J. (2011). Predicting reciprocity in social networks. In 2011 IEEE Third International Conference on Privacy, Security, Risk and Trust and 2011 IEEE Third International Conference on Social Computing, Institute of Electrical and Electronics Engineers, Piscataway, NJ, pp. 49–56.CrossRefGoogle Scholar
Durrett, R. (2019). Probability: Theory and Examples, 5th edn. Cambridge University Press.CrossRefGoogle Scholar
Java, A., Song, X., Finin, T. and Tseng, B. (2007). Why we Twitter: understanding microblogging usage and communities. In WebKDD/SNA-KDD ’07: Proceedings of the 9th WebKDD and 1st SNA-KDD 2007 workshop on Web mining and social network analysis, Association for Computing Machinery, New York, pp. 56–65.10.1145/1348549.1348556CrossRefGoogle Scholar
Jiang, B., Zhang, Z. and Towsley, D. (2015). Reciprocity in social networks with capacity constraints. In Proceedings of the 21th ACM SIGKDD International Conference on Knowledge Discovery and Data Mining (KDD ’15), Association for Computing Machinery, New York, pp. 457–466.CrossRefGoogle Scholar
Krapivsky, P. and Redner, S. (2001). Organization of growing random networks. Phys. Rev. E 63, 066123.10.1103/PhysRevE.63.066123CrossRefGoogle ScholarPubMed
Magno, G. et al. (2012). New kid on the block: exploring the Google $+$ social graph. In Proceedings of the 2012 Internet Measurement Conference (IMC ’12), Association for Computing Machinery, New York, pp. 159–170.10.1145/2398776.2398794CrossRefGoogle Scholar
Mislove, A. et al. (2007). Measurement and analysis of online social networks. In Proceedings of the 7th ACM SIGCOMM Conference on Internet Measurement (IMC ’07), Association for Computing Machinery, New York, pp. 29–42.10.1145/1298306.1298311CrossRefGoogle Scholar
Neveu, J. (1965). Mathematical Foundations of the Calculus of Probability. Holden-Day, San Francisco.Google Scholar
Newman, M., Forrest, S. and Balthrop, J. (2002). Email networks and the spread of computer viruses. Phys. Rev. E 66, 035101.10.1103/PhysRevE.66.035101CrossRefGoogle ScholarPubMed
Resnick, S. (2007). Heavy-Tail Phenomena: Probabilistic and Statistical Modeling. Springer, New York.Google Scholar
Resnick, S. and Samorodnitsky, G. (2015). Tauberian theory for multivariate regularly varying distributions with application to preferential attachment networks. Extremes 18, 349–367.CrossRefGoogle Scholar
Samorodnitsky, G. et al. (2016). Nonstandard regular variation of in-degree and out-degree in the preferential attachment model. J. Appl. Prob. 53, 146–161.CrossRefGoogle Scholar
Van der Hofstad, R. (2017). Random Graphs and Complex Networks, Vol. 1. Cambridge University Press.Google Scholar
Viswanath, B., Mislove, A., Cha, M. and Gummadi, K. (2009). On the evolution of user interaction in Facebook. In Proceedings of the 2nd ACM SIGCOMM Workshop on Social Networks (WOSN ’09), Association for Computing Machinery, New York, pp. 37–42.CrossRefGoogle Scholar
Wan, P., Wang, T., Davis, R., and Resnick, S. (2017). Fitting the linear preferential attachment model. Electron. J. Statist. 11, 3738–3780.10.1214/17-EJS1327CrossRefGoogle Scholar
Wan, P., Wang, T., Davis, R. and Resnick, S. (2020). Are extreme value estimation methods useful for network data? Extremes 23, 171–195.CrossRefGoogle Scholar
Wang, T. and Resnick, S. (2020). Degree growth rates and index estimation in a directed preferential attachment model. Stoch. Process. Appl. 130, 878–906.CrossRefGoogle Scholar
Wasserman, S. and Faust, K. (1994). Social Network Analysis: Methods and Applications. Cambridge University Press.10.1017/CBO9780511815478CrossRefGoogle Scholar