  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, Erd\H{o}s 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 sufficient conditions for P-stability,
strong stability and 8-stability. Using these sufficient 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.
