Complexity of hierarchical ensembles

TL;DR

This paper investigates the complexity of hierarchical statistical ensembles modeled as Cayley trees, showing how complexity varies with hierarchical coupling, tree branching, and time, providing explicit formulas for self-similar structures.

Contribution

It introduces a generalized combinatorial approach to measure complexity in hierarchical ensembles and derives explicit time-dependent complexity formulas for self-similar trees.

Findings

Complexity increases monotonically with hierarchical coupling and tree branching.

Variance reduces complexity when the branching exponent is below the golden mean.

Explicit time-dependent complexity formulas are derived for self-similar ensembles.

Abstract

Within the framework of generalized combinatorial approach, complexity is determined as a disorder measure for hierarchical statistical ensembles related to Cayley trees possessing arbitrary branching and number of levels. With strengthening hierarchical coupling, the complexity is shown to increase monotonically to the limit value that grows with tree branching. In contrast to the temperature dependence of thermodynamic entropy, the complexity is reduced by the variance of hierarchical statistical ensemble if the branching exponent does not exceed the gold mean. Time dependencies are found for both the probability distribution over ensemble states and the related complexity. The latter is found explicitly for self-similar ensemble and generalized for arbitrary hierarchical trees.