Complexity and the Relaxation of Hierarchical Structures

We solve exactly the problem of diffusion in an arbitrary hierarchical space. We prove that for a given "tree silhouette" $0<s<1$ the dynamic critical exponent ν ranges from s(1-s), for either uniformly or randomly multifurcating trees, to s for the most diverse ones, in qualitative agreement with a static measure of the tree's complexity. We conclude that uniform trees are optimal for information diffusion, that in thermally activated processes the temperature dependence of ν varies with the underlying tree structure, and that thin elongated trees are the only ones capable of producing a 1/f spectrum.