The phases of large networks with edge and triangle constraints

Based on numerical simulation and local stability analysis we describe the structure of the phase space of the edge/triangle model of random graphs. We support simulation evidence with mathematical proof of continuity and discontinuity for many of the phase transitions. All but one of themany phase transitions in this model break some form of symmetry, and we use this model to explore how changes in symmetry are related to discontinuities at these transitions.

💡 Research Summary

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The paper investigates the phase diagram of the edge/triangle model of dense random graphs, where two global constraints are imposed: the edge density ε and the triangle density τ. Using the graphon formalism, the authors formulate a variational principle (Theorem 2.1) that states the constrained entropy s(ε,τ) is the maximum of the graphon entropy S(g) over all symmetric measurable functions g: