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 ${\displaystyle f:V\rightarrow [0,1]}$ be a function on the vertices of $G$. Let ${\displaystyle \mu =E[f]}$ denote the mean of $f$, i.e. ${\displaystyle \mu ={\frac {1}{n}}\sum _{v\in V}f(v)}$. Then, if we let ${\displaystyle Y_{0},Y_{1},\ldots ,Y_{k}}$ 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 $\gamma >0$:

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

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

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

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 ${\displaystyle k\log n}$, whereas if we sample from an infinite family of constant-degree expanders this costs only ${\displaystyle \log n+O(k)}$. Such families exist and are efficiently constructible, e.g. the Ramanujan graphs of Lubotzky-Phillips-Sarnak.

Notes

References

• Ajtai, M.; Komlós, J.; Szemerédi, E. (1987), "Deterministic simulation in LOGSPACE", ACM, pp. 132–140, doi:10.1145/28395.28410
• Gillman, D. (1998), "A Chernoff Bound for Random Walks on Expander Graphs", SIAM Journal on Computing, Society for Industrial and Applied Mathematics, 27 (4,): 1203–1220, doi:10.1137/S0097539794268765

External links

• Proofs of the expander walk sampling theorem.