Scale-free networks with exponent one

TL;DR

This paper introduces two local rewiring models for scale-free networks with degree distribution exponent one, analyzing their properties, finite size effects, and real-world relevance.

Contribution

The paper presents novel models for equilibrium scale-free networks with exponent one, explicitly solving their properties and examining finite size effects.

Findings

Models generate uncorrelated networks in the infinite limit

Finite networks show disassortative degree correlations

Degree-dependent clustering observed in finite networks

Abstract

A majority of studied models for scale-free networks have degree distributions with exponents greater than $2$. Real networks, however, can demonstrate essentially more heavy-tailed degree distributions. We explore two models of scale-free equilibrium networks that have the degree distribution exponent $\gamma = 1$, $P(q) \sim q^{-\gamma}$. Such degree distributions can be identified in empirical data only if the mean degree of a network is sufficiently high. Our models exploit a rewiring mechanism. They are local in the sense that no knowledge of the network structure, apart from the immediate neighbourhood of the vertices, is required. These models generate uncorrelated networks in the infinite size limit, where they are solved explicitly. We investigate finite size effects by the use of simulations. We find that both models exhibit disassortative degree-degree correlations for finite network sizes. In addition, we observe a markedly degree-dependent clustering in the finite networks. We indicate a real-world network with a similar degree distribution.