Random Walks, Equidistribution and Graphical Designs

TL;DR

This paper investigates how specific initial probability measures on regular graphs can accelerate convergence of random walks to uniform distribution, with implications for graph sampling and reconstruction.

Contribution

It demonstrates that for any subset size, there exists an initial measure supported on that subset that improves convergence rates based on eigenvalues.

Findings

Existence of initial measures supported on at most ll vertices with faster convergence

Improved bounds on mixing times for specific initial distributions

Applications to graph sampling and global average reconstruction

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.