Complexity and the Relaxation of Hierarchical Structures

Abstract

We solve exactly the problem of diffusion in an arbitrary hierarchical space. We prove that for a given "tree silhouette" $0<s<\mathrm{}1\mathrm{}$ the dynamic critical exponent $\nu$ ranges from $\frac{s}{\mathrm{}(1-s)\mathrm{}}$, 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 $\nu$ varies with the underlying tree structure, and that thin elongated trees are the only ones capable of producing a $\frac{1}{f}$ spectrum.