Abstract
The friendship index measures a node’s popularity relative to its friends on a social network. The friendship index is calculated by dividing the average degree of a node’s friends by its own degree, i.e. it is the ratio of the sum of the degrees of its neighbors to the square of the degree of the node itself. Under some assumptions, the numerator of this fraction can be viewed as the sum of some random variables distributed according to the cumulative degree distribution function in the given network. It is known that for the vast majority of real complex networks, their degree distributions follow a power-law with some exponent \(\gamma \). We examine the dependence of the average value of the friendship index among nodes of the same degree k in the network on k. We will explore scale-free networks with degree-degree neutral mixing and find the limit distributions of the friendship index with the network size tending to infinity in the configuration model. Moreover, we compare our findings with the behavior of empirical friendship index distributions for several real networks.
The work was supported by the Russian Science Foundation, project 23-21-00148.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
Alipourfard, N., Nettasinghe, B., Abeliuk, A., Krishnamurthy, V., Lerman, K.: Friendship paradox biases perceptions in directed networks. Nat. Commun. 11(1), 707 (2020). https://doi.org/10.1038/s41467-020-14394-x
Bollen, J., Gonçalves, B., van de Leemput, I., Ruan, G.: The happiness paradox: your friends are happier than you. EPJ Data Sci. 6(1), 1–17 (2017). https://doi.org/10.1140/epjds/s13688-017-0100-1
Chen, N., Olvera-Cravioto, M.: Directed random graphs with given degree distributions. Stochast. Syst. 3(1), 147–186 (2013). https://doi.org/10.1214/12-SSY076
Eom, Y.H., Jo, H.H.: Generalized friendship paradox in complex networks: the case of scientific collaboration. Sci. Rep. 4, 4603 (2014). https://doi.org/10.1038/srep04603
Fotouhi, B., Momeni, N., Rabbat, M.G.: Generalized friendship paradox: an analytical approach. In: Aiello, L.M., McFarland, D. (eds.) SocInfo 2014. LNCS, vol. 8852, pp. 339–352. Springer, Cham (2015). https://doi.org/10.1007/978-3-319-15168-7_43
Higham, D.J.: Centrality-friendship paradoxes: when our friends are more important than us. J. Complex Netw. 7(4), 515–528 (2018). https://doi.org/10.1093/comnet/cny029
Hofstad, R.V.D.: Random Graphs and Complex Networks. Cambridge Series in Statistical and Probabilistic Mathematics. Cambridge University Press, Cambridge (2016). https://doi.org/10.1017/9781316779422
Jackson, M.O.: The friendship paradox and systematic biases in perceptions and social norms. J. Polit. Econ. 127(2), 777–818 (2019). https://doi.org/10.1086/701031
Lee, E., Lee, S., Eom, Y.H., Holme, P., Jo, H.H.: Impact of perception models on friendship paradox and opinion formation. Phys. Rev. E 99(5), 052302 (2019). https://doi.org/10.1103/PhysRevE.99.052302
Litvak, N., van der Hofstad, R.: Uncovering disassortativity in large scale-free networks. Phys. Rev. E 87, 022801 (2013). https://doi.org/10.1103/PhysRevE.87.022801
Momeni, N., Rabbat, M.: Qualities and inequalities in online social networks through the lens of the generalized friendship paradox. PLoS ONE 11(2), e0143633 (2016). https://doi.org/10.1371/journal.pone.0143633
Pal, S., Yu, F., Novick, Y., Bar-Noy, A.: A study on the friendship paradox – quantitative analysis and relationship with assortative mixing. Appl. Netw. Sci. 4(1), 71 (2019). https://doi.org/10.1007/s41109-019-0190-8
Paranjape, A., Benson, A.R., Leskovec, J.: Motifs in temporal networks. In: Proceedings of the Tenth ACM International Conference on Web Search and Data Mining. ACM (2017). https://doi.org/10.1145/3018661.3018731
Samorodnitsky, G., Taqqu, M.S.: Stable Non-Gaussian Random Processes. Routledge (2017). https://doi.org/10.1201/9780203738818
Sidorov, S., Mironov, S., Grigoriev, A.: Measuring the variability of local characteristics in complex networks: empirical and analytical analysis. Chaos: Interdisc. J. Nonlinear Sci. 33(6), 063106 (2023). https://doi.org/10.1063/5.0148803
Sidorov, S.P., Mironov, S.V., Grigoriev, A.A.: Friendship paradox in growth networks: analytical and empirical analysis. Appl. Netw. Sci. 6, 51 (2021). https://doi.org/10.1007/s41109-021-00391-6
Sidorov, S., Mironov, S., Malinskii, I., Kadomtsev, D.: Local degree asymmetry for preferential attachment model. In: Benito, R.M., Cherifi, C., Cherifi, H., Moro, E., Rocha, L.M., Sales-Pardo, M. (eds.) COMPLEX NETWORKS 2020 2020. SCI, vol. 944, pp. 450–461. Springer, Cham (2021). https://doi.org/10.1007/978-3-030-65351-4_36
Whitt, W.: Stochastic-Process Limits. Springer, New York (2002). https://doi.org/10.1007/b97479
Yao, D., van der Hoorn, P., Litvak, N.: Average nearest neighbor degrees in scale-free networks. Internet math. 2018, 1–38 (2018). https://doi.org/10.24166/im.02.2018
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2024 The Author(s), under exclusive license to Springer Nature Switzerland AG
About this paper
Cite this paper
Sidorov, S., Mironov, S., Grigoriev, A. (2024). Limit Distributions of Friendship Index in Scale-Free Networks. In: Ignatov, D.I., et al. Analysis of Images, Social Networks and Texts. AIST 2023. Lecture Notes in Computer Science, vol 14486. Springer, Cham. https://doi.org/10.1007/978-3-031-54534-4_23
Download citation
DOI: https://doi.org/10.1007/978-3-031-54534-4_23
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-031-54533-7
Online ISBN: 978-3-031-54534-4
eBook Packages: Computer ScienceComputer Science (R0)