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Random Walks, Equidistribution and Graphical Designs (2206.05346v1)

Published 10 Jun 2022 in math.CO, cs.DM, and math.OC

Abstract: Let $G=(V,E)$ be a $d$-regular graph on $n$ vertices and let $\mu_0$ 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 $\mu_{k+1} = A D{-1} \mu_k$, 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 $1 = \lambda_1 \geq |\lambda_2| \geq \dots \geq |\lambda_n| \geq 0$, it is well-known that the graphs for which $|\lambda_2|$ is small are those in which the random walk process converges quickly to the uniform distribution: for all initial probability measures $\mu_0$ and all $k \geq 0$, $$ \sum_{v \in V} \left| \mu_k(v) - \frac{1}{n} \right|2 \leq \lambda_2{2k}.$$ One could wonder whether this rate can be improved for specific initial probability measures $\mu_0$. We show that if $G$ is regular, then for any $1 \leq \ell \leq n$, there exists a probability measure $\mu_0$ supported on at most $\ell$ vertices so that $$ \sum_{v \in V} \left| \mu_k(v) - \frac{1}{n} \right|2 \leq \lambda_{\ell+1}{2k}.$$ The result has applications in the graph sampling problem: we show that these measures have good sampling properties for reconstructing global averages.

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