Scale-free networks with an exponent less than two

TL;DR

This paper investigates scale-free networks with degree distribution exponent less than two, revealing they have unique properties such as rapid link growth, small-world characteristics, and finite clustering, supported by a simple analytical model.

Contribution

It introduces a simple prototype model for such networks, providing detailed analytical insights into their distinct properties compared to networks with higher exponents.

Findings

Number of links grows faster than number of nodes

Networks exhibit small world property with logarithmic diameter growth

Clustering coefficient remains finite

Abstract

We study scale free simple graphs with an exponent of the degree distribution $\gamma$ less than two. Generically one expects such extremely skewed networks -- which occur very frequently in systems of virtually or logically connected units -- to have different properties than those of scale free networks with $\gamma>2$: The number of links grows faster than the number of nodes and they naturally posses the small world property, because the diameter increases by the logarithm of the size of the network and the clustering coefficient is finite. We discuss a simple prototype model of such networks, inspired by real world phenomena, which exhibits these properties and allows for a detailed analytical investigation.