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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]
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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
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Copyright
© The Author(s), 2022. Published by Cambridge University Press on behalf of Applied Probability Trust

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