Remarks on Lecture 3

Today we saw the connection between mixing times of random walks and the second eigenvalue of the normalized Laplacian. Recall that {G=(V,E,w)} is an {n} -vertex, weighted graph.

First, we introduced the lazy walk matrix,

\displaystyle  W_G = \frac{1}{2} (I + A_G D_G^{-1})\,,

which is related to the normalized Laplacian via,

\displaystyle  W_G = I - \frac{1}{2} D_G^{1/2} \mathcal{L}_G D_G^{-1/2}\,.

In particular, if we arrange the eigenvalues of {W_G} as {1=\beta_1 \geq \beta_2 \geq \cdots \geq \beta_n} , then

\displaystyle  \beta_i = 1 - \frac{\lambda_i}{2}\,,

where {\lambda_i} is the {i} th smallest eigenvalue of {\mathcal{L}_G} . Furthermore, the eigenfunctions of {W_G} are given by {g = D_G^{1/2} f} , where {f} is an eigenfunction of {\mathcal L_G} .

The eigenfunctions of {W_G} are orthogonal with respect to the inner product

\displaystyle  (f,g) = \sum_{u \in V} \frac{1}{w(u)} f(u) g(u)\,.

We will refer to this Hilbert space as {\ell^2(1/w)} . Recall that {\|f\|_{\ell^2(1/w)} = \sqrt{(f,f)} = \sqrt{\sum_{v \in V} \frac{1}{w(v)} f(v)^2}} .

If we define the stationary distribution {\pi : V \rightarrow [0,1]} by

\displaystyle  \pi(u) = \frac{w(u)}{w(V)} = \frac{w(u)}{2 w(E)},

then {W_G \pi = \pi} .

Let {f_1, f_2, \ldots, f_n} be an {\ell^2(1/w)} -orthornormal system of eigenvectors for {W_G} . Now fix an initial distribution {p_0 : V \rightarrow [0,1]} and write,

\displaystyle  p_0 = \sum_{i=1}^n \alpha_i f_i\,.

Then we have,

\displaystyle  p_t = W^t p_0 = \pi + \sum_{i > 1} \alpha_i \left(1-\frac{\lambda_i}{2}\right)^t f_i\,.

In particular,

\displaystyle  \|p_t - \pi\|_{\ell^2(1/w)}^2 = \sum_{i > 1} \alpha_i^2 \left(1-\frac{\lambda_i}{2}\right)^{2t} \leq \left(1-\frac{\lambda_2}{2}\right)^{2t} \sum_{i > 1} \alpha_i^2 \leq \left(1-\frac{\lambda_2}{2}\right)^{2t} \|p_0\|_{\ell^2(1/w)}^2\,.

Let {\pi_* = \min_{u \in V} \pi(u)} be the minimum stationary measure. A particularly strongly notion of mixing time is given by the following definition.

\displaystyle  \tau_{\infty} = \sup_{p_0} \min \left\{ t : \|p_t - \pi\|_{\infty} < \frac{\pi_*}{4} \right\}\,

where the supremum is taken over all initial distributions {p_0} . By time {\tau_{\infty}} , the distribution {p_t} is pointwise very close to the stationary distribution {\pi} .

Let {w^* = \max_{u \in V} w(u)} and {w_* = \min_{u \in V} w(u)} . Using our preceding calculation, we have

\displaystyle  \|p_t - \pi\|_{\infty} \leq \sqrt{w^*} \|p_t - \pi\|_{\ell^2(1/w)} \leq \sqrt{w^*} \left(1-\frac{\lambda_2}{2}\right)^{t} \|p_0\|_{\ell^2(V,w)} \leq \left(1-\frac{\lambda_2}{2}\right)^{t} \sqrt{\frac{w^*}{w_*}}\,.

Observe that {\frac{w^*}{w_*} \leq \frac{1}{\pi_*}} , so if we now choose {t \asymp \frac{1}{\lambda_2} \log \frac{1}{\pi_*}} appropriately, then

\displaystyle  \|p_t - \pi\|_{\infty} < \frac{\pi_*}{4},

yielding our main result:

\displaystyle  \tau_{\infty} \lesssim \frac{1}{\lambda_2} \log \frac{1}{\pi_*}\,.

One can also proof that {\tau_{\infty} \gtrsim \frac{1}{\lambda_2}} (see the exercises). In general, we will be concerned with weaker notions of mixing. We’ll discuss those when they arise.


Prove that {\tau_{\infty} \gtrsim \frac{1}{\lambda_2}} for any graph G.


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