Towards a Theory of Scale-Free Graphs: Definition, Properties, and Implications (Extended Version)

TL;DR

This paper introduces a precise mathematical definition of scale-free graphs, proves properties consistent with existing claims, and clarifies misconceptions in the scale-free literature through rigorous analysis.

Contribution

It provides a new formal definition of scale-free graphs and establishes foundational results, advancing the theoretical understanding of their properties.

Findings

Proposes a rigorous, structural definition of scale-free graphs.

Shows scale-free graphs are likely outcomes of random processes.

Clarifies misconceptions and contradictions in existing literature.

Abstract

Although the ``scale-free'' literature is large and growing, it gives neither a precise definition of scale-free graphs nor rigorous proofs of many of their claimed properties. In fact, it is easily shown that the existing theory has many inherent contradictions and verifiably false claims. In this paper, we propose a new, mathematically precise, and structural definition of the extent to which a graph is scale-free, and prove a series of results that recover many of the claimed properties while suggesting the potential for a rich and interesting theory. With this definition, scale-free (or its opposite, scale-rich) is closely related to other structural graph properties such as various notions of self-similarity (or respectively, self-dissimilarity). Scale-free graphs are also shown to be the likely outcome of random construction processes, consistent with the heuristic definitions implicit in existing random graph approaches. Our approach clarifies much of the confusion surrounding the sensational qualitative claims in the scale-free literature, and offers rigorous and quantitative alternatives.