The switch chain is a well-studied Markov chain which can be used to sample approximately uniformly from the set $\Omega(\boldsymbol{d})$ of all graphs with a given degree sequence $\boldsymbol{d}$. Polynomial mixing time (rapid mixing) has been established for the switch chain under various conditions on the degree sequences. Amanatidis and Kleer introduced the notion of strongly stable families of degree sequences, and proved that the switch chain is rapidly mixing for any degree sequence from a strongly stable family. Using a different approach, Erdős et al. recently extended this result to the (possibly larger) class of P-stable degree sequences, introduced by Jerrum and Sinclair in 1990. We define a new notion of stability for a given degree sequence, namely $k$-\emph{stability}, and prove that if a degree sequence $\boldsymbol{d}$ is 8-stable then the switch chain on $\Omega(\boldsymbol{d})$ is rapidly mixing. We also provide necessary conditions for P-stability, strong stability and 8-stability. Using these necessary conditions, we give the first proof of P-stability for various families of heavy-tailed degree sequences, including power-law degree sequences, and show that the switch chain is rapidly mixing for these families. We further extend these notions and results to directed degree sequences.

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切换马尔可夫链与稳定度序列的混合时间

开关链是一个经过充分研究的马尔可夫链，可用于从具有给定度数序列 $\boldsymbol{d}$ 的所有图的集合 $\Omega(\boldsymbol{d})$ 中近似均匀地采样。多项式混合时间（快速混合）已经在度数序列的各种条件下为开关链建立。Amanatidis 和 Kleer 引入了度序列的强稳定族的概念，并证明了开关链正在快速混合来自强稳定族的任何度序列。使用不同的方法，Erdős 等人。最近将这个结果扩展到（可能更大）类 P 稳定度序列，由 Jerrum 和 Sinclair 在 1990 年引入。我们为给定度序列定义了一个新的稳定性概念，即 $k$-\emph{stability}，并证明如果度数序列 $\boldsymbol{d}$ 是 8 稳定的，那么 $\Omega(\boldsymbol{d})$ 上的开关链正在快速混合。我们还为P-稳定、强稳定和8-稳定提供了必要条件。使用这些必要条件，我们首次证明了各种重尾度序列家族（包括幂律度序列）的 P 稳定性，并表明这些家族的开关链正在快速混合。我们进一步将这些概念和结果扩展到有向度序列。并表明这些家庭的开关链正在迅速混合。我们进一步将这些概念和结果扩展到有向度序列。并表明这些家庭的开关链正在迅速混合。我们进一步将这些概念和结果扩展到有向度序列。