Conductance of a Markov chain ($\Phi (A)$)

For a reversible Markov chain with stationary flow $Q(x,y)=\pi (x)P(x,y)$, the conductance of a set $A$ isIts Cheeger constant minimizes the boundary flow divided by the smaller stationary mass of the two sides of the cut.

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 isThis 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 $k\le n/2$ 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.^[ref]