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.

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.