Creation probabilities of hierarchical trees

TL;DR

This paper investigates the probabilistic conditions for creating various hierarchical trees, establishing a connection between level creation probabilities and their integration into a unified structure, utilizing deformed algebra for a consistent framework.

Contribution

It introduces a comprehensive probabilistic model for hierarchical tree creation, incorporating deformed algebra, and provides analytical and numerical results for different tree types including Fibonacci and free-scale trees.

Findings

Derived a general expression for tree creation probability

Established a link between level probabilities and overall structure formation

Numerically determined probabilities for Fibonacci and free-scale trees

Abstract

We consider both analytically and numerically creation conditions of diverse hierarchical trees. A connection between the probabilities to create hierarchical levels and the probability to associate these levels into united structure is found. We argue a consistent probabilistic picture requires making use of the deformed algebra. Our consideration is based on study of main types of hierarchical trees, among which both regular and degenerate ones are studied analytically, while the creation probabilities of the Fibonacci and free-scale trees are determined numerically. We find a general expression for the creation probability of an arbitrary tree and calculate the sum of terms of deformed geometrical progression that appears at consideration of the degenerate tree.