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

Abstract

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 ∝ n ^γ, 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.