### Abstract

Sampling uniform simple graphs with power-law degree distributions with degree exponent $\tau\in(2,3)$ is a non-trivial problem. We propose a method to sample uniform simple graphs that uses a constrained version of the configuration model together with a Markov Chain switching method. We test the convergence of this algorithm numerically in the context of the presence of small subgraphs. We then compare the number of triangles in uniform random graphs with the number of triangles in the erased configuration model. Using simulations and heuristic arguments, we conjecture that the number of triangles in the erased configuration model is larger than the number of triangles in the uniform random graph, provided that the graph is sufficiently large.

Original language | English |
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Article number | cny013 |

Pages (from-to) | 210-225 |

Number of pages | 16 |

Journal | Journal of Complex Networks |

Volume | 7 |

Issue number | 2 |

Early online date | 16 Aug 2018 |

DOIs | |

Publication status | Published - 1 Apr 2019 |

Externally published | Yes |

### Keywords

- math.PR
- physics.soc-ph

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## Cite this

Bannink, T., Hofstad, R. V. D., & Stegehuis, C. (2019). Switch chain mixing times and triangle counts in simple random graphs with given degrees.

*Journal of Complex Networks*,*7*(2), 210-225. [cny013]. https://doi.org/10.1093/comnet/cny013