Measuring the hierarchy of feedforward networks

Area of Science:

• Complex systems analysis
• Network science
• Information theory

Background:

• Quantifying hierarchy in complex systems is challenging.
• Existing methods lack a unified framework for hierarchical structures.
• Ordered systems exhibit properties like predictability and pyramidal organization.

Purpose of the Study:

• To define and quantify hierarchy in ordered structures.
• To develop a novel hierarchical index based on graph and information theory.
• To differentiate between hierarchical, antihierarchical, and nonhierarchical systems.

Main Methods:

• Defined three conditions for hierarchical structures: order, predictability, and pyramidal structure.
• Developed a hierarchical index using concepts from graph theory and information theory.
• Utilized two entropies (onward flow and backward reversion) to balance predictability and pyramidality.

Main Results:

• The hierarchical index quantifies the hierarchical character of systems representable as feedforward causal graphs (directed acyclic graphs).
• Maximal hierarchical systems correspond to feedforward trees, while maximal antihierarchical systems correspond to inverted tree graphs.
• Nonhierarchical systems (e.g., linear chains, fully connected feedforward graphs) yield null values for the index due to lack of pyramid structure or predictability.

Conclusions:

• The proposed hierarchical index provides a robust method for classifying complex systems based on their structural hierarchy.
• The framework successfully distinguishes between systems with varying degrees of order, predictability, and pyramidal organization.
• This work offers a theoretical foundation for analyzing hierarchical properties in diverse scientific domains.