# Switch chain mixing times and triangle counts in simple random graphs with given degrees

Tom Bannink, Remco van der Hofstad, Clara Stegehuis

3 Citations (Scopus)

## 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 cny013 210-225 16 Journal of Complex Networks 7 2 16 Aug 2018 https://doi.org/10.1093/comnet/cny013 Published - 1 Apr 2019 Yes

## Keywords

• math.PR
• physics.soc-ph