The phases of large networks with edge and triangle constraints

Abstract

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.

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• Finite-size effects in exponential random graphs and cluster evaporation
• Ensemble inequivalence and phase transitions in unlabeled networks
• Regular graphs with linearly many triangles
• Nucleation during phase transitions in random networks
• Ground States for Exponential Random Graphs
• Recovering Nonuniform Planted Partitions via Iterated Projection