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Typical large graphs with given edge and triangle densities
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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.
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Emergence in graphs with near-extreme constraints
Near the boundary of feasible edge and triangle densities, entropy-optimal graphons are unique, multipodal, and analytic in the constraints, yielding infinitely many phases.
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