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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}
}

. 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… Expand

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.Expand

The book Large Networks and Graph Limits, xiv + 475 pp., published in late 2012, comprises five parts, the first an illuminating introduction and the last a tantalizing taste of how the scope of the… Expand

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… Expand

The most famous result of extremal combinatorics is probably the celebrated theorem of Turán [20] determining the maximal number ex(n;Kr) of edges in a Kr-free graph on n vertices. Asymptotically,… Expand

It is proved that under constraints graphs are “multipodal”: asymptotically in the number of vertices there is a partition of the vertices into a set of well-defined probabilities and the nonuniqueness extends to entropy maximizers in the interior of the phase space.Expand

The first rigorous proofs of ``degeneracy'' observed in these models are given, showing that for many models, the extra sufficient statistics are useless: most realizations look like the results of a simple Erd\H{o}s-R\'{e}nyi model.Expand