Remarks on power-law random graphs

TL;DR

This paper explores a power-law random graph model within the graphon framework, analyzing its structural properties in sub-critical and super-critical regimes and revealing phase transition behaviors.

Contribution

It introduces a graphon-based analysis of power-law random graphs, detailing their limit structures in different regimes and identifying boundary cases of graph convergence.

Findings

In the sub-critical regime, the graph is empty with high probability.

In the rare case of non-empty graphs, only a single edge exists.

In the super-critical regime, a non-trivial limit graph emerges.

Abstract

The theory of graphons is an important tool in understanding properties of large networks. We investigate a power-law random graph model and cast it in the graphon framework. The distinctively different structures of the limit graph are explored in detail in the sub-critical and super-critical regimes. In the sub-critical regime, the graph is empty with high probability, and in the rare event that it is non-empty, it consists of a single edge. Contrarily, in the super-critical regime, a non-trivial random graph exists in the limit, and it serves as an uncovered boundary case between different types of graph convergence.