Taylor’s power law for the &lt;italic&gt;N&lt;/italic&gt;-stars network evolution model

István Fazekas,Csaba Noszály,Noémi Uzonyi

doi:10.15559/19-vmsta137

Abstract

Taylor's power law states that the variance function decays as a power law. It is observed for population densities of species in ecology. For random networks another power law, that is, the power law degree distribution is widely studied. In this paper the original Taylor's power law is considered for random networks. A precise mathematical proof is presented that Taylor's power law is asymptotically true for the $N$-stars network evolution model.

Highlights

Taylor’s power law is a well-known empirical pattern in ecology
Is the original Taylor’s power law true for random networks? First we considered data sets of real life networks
In this paper we prove an asymptotic Taylor’s power law for the N -stars network evolution model

Summary

Introduction

Taylor’s power law is a well-known empirical pattern in ecology. Its general form is. There are papers studying Taylor’s power law on networks A rigorous definition of the preferential attachment model was given in [4], where a mathematical proof of the power law degree distribution was presented. At each time step either a new vertex or an old one generates new edges In both cases the terminal vertices can be chosen either uniformly or according to the preferential attachment rule. We encountered the following more specific problem: Find network structures where Taylor’s power law is true. To this end we analysed the above N-stars network evolution model. In this paper we prove an asymptotic Taylor’s power law for the N -stars network evolution model.

The N-stars network evolution model and the main results

Numerical results