The triangle switch Markov chain is designed to generate random graphs with given degree sequence, but having more triangles than would appear under the uniform distribution. Transition probabilities of the chain depends on a parameter, called the activity, which is used to assign higher stationary probability to graphs with more triangles. In previous work we proved ergodicity of the triangle switch chain for regular graphs. Here we prove ergodicity for all sequences with minimum degree at least 3, and show rapid mixing of the chain when the activity and the maximum degree are not too large. As far as we are aware, this is the first rigorous analysis of a Markov chain algorithm for generating graphs from a a known non-uniform distribution.

翻译：三角开关 Markov 链的设计是生成带有给定度序列的随机图表, 但其三角形比统一分布下显示的要多。 链的过渡概率取决于参数, 称为活动, 用于给带有更多三角形的图形分配更高的固定概率。 在先前的工作中, 我们证明三角开关链对普通图形具有任意性。 在这里, 我们证明所有序列至少具有最小度 3, 并且显示当活动和最大度不太大时链的快速混合 。 据我们所知, 这是第一次严格分析 Markov 链算法, 用于从已知的非统一分布中生成图形 。