Random Walks, Equidistribution and Graphical Designs
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Let G=(V,E) be a d-regular graph on n vertices and let μ_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 μ_k+1 = A D^-1μ_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 = λ_1 ≥ |λ_2| ≥…≥ |λ_n| ≥ 0, it is well-known that the graphs for which |λ_2| is small are those in which the random walk process converges quickly to the uniform distribution: for all initial probability measures μ_0 and all k ≥ 0, ∑_v ∈ V| μ_k(v) - 1/n|^2 ≤λ_2^2k. One could wonder whether this rate can be improved for specific initial probability measures μ_0. We show that if G is regular, then for any 1 ≤ℓ≤ n, there exists a probability measure μ_0 supported on at most ℓ vertices so that ∑_v ∈ V| μ_k(v) - 1/n|^2 ≤λ_ℓ+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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