On the Diffusion Geometry of Graph Laplacians and Applications
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We study directed, weighted graphs G=(V,E) and consider the (not necessarily symmetric) averaging operator (Lu)(i) = -∑_j ∼_ ip_ij (u(j) - u(i)), where p_ij are normalized edge weights. Given a vertex i ∈ V, we define the diffusion distance to a set B ⊂ V as the smallest number of steps d_B(i) ∈N required for half of all random walks started in i and moving randomly with respect to the weights p_ij to visit B within d_B(i) steps. Our main result is that the eigenfunctions interact nicely with this notion of distance. In particular, if u satisfies Lu = λ u on V and B = { i ∈ V: - ε≤ u(i) ≤ε}≠∅, then, for all i ∈ V, d_B(i) ( 1/|1-λ|) ≥( |u(i)| /u_L^∞) - (1/2 + ε). d_B(i) is a remarkably good approximation of |u| in the sense of having very high correlation. The result implies that the classical one-dimensional spectral embedding preserves particular aspects of geometry in the presence of clustered data. We also give a continuous variant of the result which has a connection to the hot spots conjecture.
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