Random Walks, Equidistribution and Graphical Designs

Let $G=(V,E)$ be a $d$-regular graph on $n$ vertices and let $\mu0$ be a probability measure on $V$. The act of moving to a randomly chosen neighbor leads to a sequence of probability measures supported on $V$ given by $\mu{k+1} = A D^{-1} \muk$, where $A$ is the adjacency matrix and $D$ is the diagonal matrix of vertex degrees of $G$. Ordering the eigenvalues of $ A D^{-1}$ as $1 = \lambda1 \geq |\lambda2| \geq \dots \geq |\lambdan| \geq 0$, it is well-known that the graphs for which $|\lambda2|$ is small are those in which the random walk process converges quickly to the uniform distribution: for all initial probability measures $\mu0$ and all $k \geq 0$, $$ \sum{v \in V} \left| \muk(v) - \frac{1}{n} \right|² \leq \lambda2^{2k}.$$ One could wonder whether this rate can be improved for specific initial probability measures $\mu0$. We show that if $G$ is regular, then for any $1 \leq \ell \leq n$, there exists a probability measure $\mu0$ supported on at most $\ell$ vertices so that $$ \sum{v \in V} \left| \muk(v) - \frac{1}{n} \right|² \leq \lambda{\ell+1}^{2k}.$$ The result has applications in the graph sampling problem: we show that these measures have good sampling properties for reconstructing global averages.