Typical large graphs with given edge and triangle densities

TL;DR

This paper investigates the typical structure of large graphs with fixed edge and triangle densities, revealing sharp phase transitions and singularities using large deviation principles, and extends the analysis to other cycle densities.

Contribution

It extends the analysis of large graphs to include typical structures at fixed intermediate densities, identifying phase transitions and singularities with rigorous proofs.

Findings

Typical graphs show sharp singularities as densities vary.

The nature of singularities is precisely characterized.

Extension to graphs with fixed densities of other cycles is straightforward.

Abstract

The analysis of large simple graphs with extreme values of the densities of edges and triangles has been extended to the statistical structure of typical graphs of fixed intermediate densities, by the use of large deviations of Erdoes-Renyi graphs. We prove that the typical graph exhibits sharp singularities as the constraining densities vary between different curves of extreme values, and we determine the precise nature of the singularities. The extension to graphs with fixed densities of edges and k-cycles for odd k>3 is straightforward and we note the simple changes in the proof.