# 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} }

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 Erdős-Rényi 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.

## References

SHOWING 1-10 OF 29 REFERENCES

Singularities in the Entropy of Asymptotically Large Simple Graphs

- Mathematics
- 2015

We prove that the asymptotic entropy of large simple graphs, as a function of fixed edge and triangle densities, is nondifferentiable along a certain curve. We also determine the precise…

Bipodal structure in oversaturated random graphs

- Mathematics, Computer ScienceArXiv
- 2015

It is proved that, for all but finitely many values of the edge density, the typical large graph is bipodal with parameters varying analytically with the densities.

Asymptotic Structure of Graphs with the Minimum Number of Triangles

- Mathematics, Computer ScienceCombinatorics, Probability and Computing
- 2016

This work describes the asymptotic structure of extremal graphs by characterizing the set of flag algebra homomorphisms that minimize the triangle density by considering the problem of minimizing the number of triangles in a graph of given order and size.

The phases of large networks with edge and triangle constraints

- Mathematics, PhysicsArXiv
- 2017

Based on numerical simulation and local stability analysis, the structure of the phase space of the edge/triangle model of random graphs is described, and changes in symmetry are related to discontinuities at these transitions.

Limits of dense graph sequences

- Computer Science, MathematicsJ. Comb. Theory, Ser. B
- 2006

We show that if a sequence of dense graphs G"n has the property that for every fixed graph F, the density of copies of F in G"n tends to a limit, then there is a natural ''limit object,'' namely a…

Nonlinear large deviations

- Mathematics
- 2014

We present a general technique for computing large deviations of nonlinear functions of independent Bernoulli random variables. The method is applied to compute the large deviation rate functions for…

The large deviation principle for the Erdős-Rényi random graph

- Computer Science, MathematicsEur. J. Comb.
- 2011

The formulation and proof of the main result uses the recent development of the theory of graph limits by Lovasz and coauthors and Szemeredi's regularity lemma from graph theory to establish a large deviation principle under an appropriate topology.

Phase transitions in a complex network

- Physics, Mathematics
- 2013

We study a mean field model of a complex network, focusing on edge and triangle densities. Our first result is the derivation of a variational characterization of the entropy density, compatible with…

Finitely forcible graphons

- Computer Science, MathematicsJ. Comb. Theory, Ser. B
- 2011

The main focus is the case when the family contains only one element, i.e., a unique structure is forced by finitely many subgraph densities, which implies that finitely forcible graphons are ''rare'', and exhibit simple and explicit non-forcible graphons.

AN EXTREMAL PROBLEM IN GRAPH THEORY

- 2001

G(?z; I) will denote a graph of n vertices and 1 edges. Let fO(lz, K) be the smallest integer such that there is a G (n; f,, (n, k)) in which for every set of K vertices there is a vertex joined to…