Measuring the Hierarchy of Feedforward Networks
Bernat Corominas-Murtra, Joaqu\'in Go\~ni, Carlos Rodr\'iguez-Caso,, Ricard Sol\'e

TL;DR
This paper introduces a quantitative hierarchical index for feedforward networks based on graph and information theory, enabling classification of structures as hierarchical, anti-hierarchical, or non-hierarchical.
Contribution
It defines a novel hierarchical index using entropy measures to quantify hierarchy in directed acyclic graphs, distinguishing different causal structures.
Findings
Maximal hierarchy in feedforward trees and inverted trees identified.
Null hierarchy values correspond to linear chains and fully connected graphs.
The index effectively classifies various causal network structures.
Abstract
In this paper we explore the concept of hierarchy as a quantifiable descriptor of ordered structures, departing from the definition of three conditions to be satisfied for a hierarchical structure: {\em order}, {\em predictability} and {\em pyramidal structure}. According to these principles we define a hierarchical index taking concepts from graph and information theory. This estimator allows to quantify the hierarchical character of any system susceptible to be abstracted in a feedforward causal graph, i.e., a directed acyclic graph defined in a single connected structure. Our hierarchical index is a balance between this predictability and pyramidal condition by the definition of two entropies: one attending the onward flow and other for the backward reversion. We show how this index allows to identify hierarchical, anti-hierarchical and non hierarchical structures. Our formalism…
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