Expander walk sampling

In the mathematical discipline of graph theory, the expander walk sampling theorem states that sampling vertices in an expander graph by doing a random walk is almost as good as sampling the vertices independently from a uniform distribution. The earliest version of this theorem is due to Ajtai, Komlós & Szemerédi (1987), and the more general version is typically attributed to Gillman (1998).

Statement

Let $G=(V,E)$ be an expander graph with normalized second-largest eigenvalue $\lambda$. Let $n$ denote the number of vertices in $G$. Let $f:V\rightarrow [0,1]$ be a function on the vertices of $G$. Let $\mu =E[f]$ denote the mean of $f$, i.e. $\mu ={\frac {1}{n}}\sum _{v\in V}f(v)$. Then, if we let $Y_{0},Y_{1},\ldots ,Y_{k}$ denote the vertices encountered in a $k$-step random walk on $G$ starting at a random vertex $Y_{0}$, we have the following for all $\gamma >0$:

$\Pr \left[{\frac {1}{k}}\sum _{i=0}^{k}f(Y_{i})-\mu >\gamma \right]\leq e^{-\Omega (\gamma ^{2}(1-\lambda )k)}.$

Here the $\Omega$ hides an absolute constant $\geq 1/10$. An identical bound holds in the other direction:

$\Pr \left[{\frac {1}{k}}\sum _{i=0}^{k}f(Y_{i})-\mu <-\gamma \right]\leq e^{-\Omega (\gamma ^{2}(1-\lambda )k)}.$

Uses

This theorem is useful in randomness reduction in the study of derandomization. Sampling from an expander walk is an example of a randomness-efficient sampler. Note that the number of bits used in sampling $k$ independent samples from $f$ is $k\log n$, whereas if we sample from an infinite family of constant-degree expanders this costs only $\log n+O(k)$. Such families exist and are efficiently constructible, e.g. the Ramanujan graphs of Lubotzky-Phillips-Sarnak.

References

• Ajtai, M.; Komlós, J.; Szemerédi, E. (1987). "Deterministic simulation in LOGSPACE". Proceedings of the nineteenth annual ACM conference on Theory of computing - STOC '87. pp. 132–140. doi:10.1145/28395.28410. ISBN 0897912217. {{cite book}}: |work= ignored (help); Unknown parameter |booktitle= ignored (help)
• Gillman, D. (1998), "A Chernoff Bound for Random Walks on Expander Graphs", SIAM Journal on Computing, 27 (4), Society for Industrial and Applied Mathematics: 1203–1220, doi:10.1137/S0097539794268765, S2CID 26319459

External links

• Proofs of the expander walk sampling theorem. [1] [2]