From Quasirandom graphs to Graph Limits and Graphlets

TL;DR

This paper extends the concept of quasirandom graphs to graph limits called graphlets, establishing equivalences between spectral and cut distance convergence, and characterizing low-rank graphlets in dense and sparse graphs.

Contribution

It introduces the notion of graphlets as a new form of graph limit and proves their properties, including convergence equivalences and low-rank characterizations.

Findings

Spectral and cut distance convergence are equivalent for graph sequences.

Graphlets can be characterized by low-rank structures.

The framework applies to both dense and sparse graphs.

Abstract

We generalize the notion of quasirandom which concerns a class of equivalent properties that random graphs satisfy. We show that the convergence of a graph sequence under the spectral distance is equivalent to the convergence using the (normalized) cut distance. The resulting graph limit is called graphlets. We then consider several families of graphlets and, in particular, we characterize graphlets with low ranks for both dense and sparse graphs.