Cheeger bound

In mathematics, the Cheeger bound is a bound of the second largest eigenvalue of the transition matrix of a finite-state, discrete-time, reversible stationary Markov chain. It can be seen as a special case of Cheeger inequalities in expander graphs.

Let $$X$$ be a finite set and let $$K(x,y)$$ be the transition probability for a reversible Markov chain on $$X$$. Assume this chain has stationary distribution $$\pi$$.

Define


 * $$Q(x,y) = \pi(x) K(x,y) $$

and for $$A,B \subset X $$ define


 * $$Q(A \times B) = \sum_{x \in A, y \in B} Q(x,y). $$

Define the constant $$\Phi$$ as


 * $$ \Phi = \min_{S \subset X, \pi(S) \leq \frac{1}{2}} \frac{Q (S \times S^c)}{\pi(S)}. $$

The operator $$K,$$ acting on the space of functions from $$|X|$$ to $$\mathbb{R}$$, defined by


 * $$ (K \phi)(x) = \sum_y K(x,y) \phi(y) $$

has eigenvalues $$ \lambda_1 \geq \lambda_2 \geq \cdots \geq \lambda_n $$. It is known that $$\lambda_1 = 1$$. The Cheeger bound is a bound on the second largest eigenvalue $$\lambda_2$$.

 Theorem (Cheeger bound):


 * $$ 1 - 2 \Phi \leq \lambda_2 \leq 1 - \frac{\Phi^2}{2}. $$