An Allometric Scaling for the Number of Representative Nodes in Social Networks

  • Liang ZhaoEmail author
  • Tianyi Peng
Conference paper
Part of the Springer Proceedings in Complexity book series (SPCOM)


This paper studies the scale of the size of a representative node set in social networks. First, a simple distance-based representative model is proposed. Then, with two small-world like assumptions which are widely observed in large-scale online social networks, it is shown that the size R of such a representative set satisfies an allometric scaling R ∝ nγ, where n is the size of the network and γ is a constant such that 0 ≤ γ < 1. In particular, a theoretical analysis using Dunbar’s Number as the average degree of nodes suggests 1∕3 ≤ γ ≤ 5∕9 for large-scale real social networks. This is the first theoretical model that can explain the phenomenon that the number of congressional representatives scales to the \(\frac {2}{5}\)-th power (i.e., γ = 2∕5) of the population in real world. It also suggests that, in order to represent (or to influence) a majority in a social network, a surprisingly small (sublinear) number of representatives is enough. For instance, the number is a few thousands for Facebook which has more than two billions users. This demonstrates how easy to spread information in social networks.


Social network Representative nodes Allometric scaling Influence Congressional representative Pyramid structure Democratic number 



This research was supported by JSPS KAKENHI Grant Number 18K11182.


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© Springer Nature Switzerland AG 2020

Authors and Affiliations

  1. 1.Graduate School of Advanced Integrated Studies in Human Survivability (Shishu-Kan)Kyoto UniversityKyotoJapan
  2. 2.Lab for Information and Decision SystemsMassachusetts Institute of TechnologyCambridgeUSA

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