# The phases of large networks with edge and triangle constraints

## TL;DR

This paper investigates the phase space structure of a random graph model constrained by edges and triangles, revealing symmetry-breaking transitions supported by simulations and mathematical proofs.

## Contribution

It provides a detailed analysis of phase transitions in the edge/triangle model, including proofs of continuity and discontinuity, and explores symmetry-breaking phenomena.

## Key findings

- Most phase transitions involve symmetry breaking
- Mathematical proofs support simulation results
- Identifies conditions for continuous and discontinuous transitions

## 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.