= Conductance profile
The <conductance profile> records bottlenecks at different stationary sizes in a finite <irreducible Markov chain> with at least two states. With <stationary distribution> $\pi$ and stationary flow $Q(x,y)=\pi(x)P(x,y)$, one convention is
$$
\Phi(r)=\min_{0<\pi(A)\leq r}\frac{Q(A,A^c)}{\pi(A)},\qquad \pi_{\min}\leq r\leq\tfrac12.
$$
This domain ensures that the minimizing family is nonempty. Other normalizations use the smaller side of a cut, so a convention must accompany a quoted profile.
For a lazy nearest-neighbor walk on an $n$-cycle, holding with <probability> $1/2$, a consecutive block of $k\leq n/2$ vertices has boundary flow $1/(2n)$ and stationary mass $k/n$, hence conductance $1/(2k)$. A profile therefore reveals how bottlenecks change with set size rather than retaining only one <global minimum>.<Mixing Time Bounds via the Spectral Profile by Sharad Goel and others>[Section 2.2, conductance-profile comparison; cycle calculation supplied here]{ref}
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