Two Power Series Models of Self-Similarity in Social Networks

TL;DR

This paper introduces two power series models to describe self-similarity in social networks, comparing them to Zipf and Benford distributions, and explores their implications for network evolution and organization.

Contribution

It proposes novel power series models with self-similarity properties, including a model where the scaling factor is the golden ratio, and analyzes network evolution through oscillations between measures.

Findings

One model uses geometric mean for the middle term, contrasting with harmonic mean in Zipf.

The scaling factor at one level is identified as the golden ratio.

A model describes network evolution via oscillations between two self-similarity measures.

Abstract

Two power series models are proposed to represent self-similarity and they are compared to the Zipf and Benford distributions. Since evolution of a social network is associated with replicating self-similarity at many levels, the nature of interconnections can serve as a measure of the optimality of its organization. In contrast with the Zipf distribution where the middle term is the harmonic mean of the adjoining terms, our distribution considers the middle term to be the geometric mean. In one of the power series models, the scaling factor at one level is shown to be the golden ratio. A model for evolution of networks by oscillations between two different self-similarity measures is described.