• Corpus ID: 239998123

Typical large graphs with given edge and triangle densities

@inproceedings{Neeman2021TypicalLG,
  title={Typical large graphs with given edge and triangle densities},
  author={Joe Neeman and Charles Radin and Lorenzo A Sadun},
  year={2021}
}

• Joe Neeman, C. Radin, L. Sadun
• Published 26 October 2021
• Mathematics

. 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 inter-mediate densities, by the use of large deviations of Erd˝os-R´enyi 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 ex-tension 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. asymptotic probabilistic structure, and use the rate function for certain large deviations of Erd˝os-R´enyi graphs to interpret typicality, a notion central to our analysis. We give a careful discussion of typicality in Section 2.1, but informally it means ‘all but exponentially few’ graphs with the given density constraints, exponential in the number of vertices of the system.