Hierarchy Measures in Complex Networks

TL;DR

This paper introduces a measure of hierarchy in complex networks based on node degrees, analyzes how hierarchy varies with network parameters, and compares real-world networks to extremal and random models to understand their hierarchical and modular features.

Contribution

It proposes a simple dynamical process to construct networks with maximal or minimal hierarchy and analyzes how hierarchy depends on the degree distribution exponent in scale-free networks.

Findings

Hierarchy measure declines with increasing degree distribution exponent

Maximum hierarchy occurs for exponent ≤ 2

Hierarchy approaches zero for exponent > 3

Abstract

Using each node's degree as a proxy for its importance, the topological hierarchy of a complex network is introduced and quantified. We propose a simple dynamical process used to construct networks which are either maximally or minimally hierarchical. Comparison with these extremal cases as well as with random scale-free networks allows us to better understand hierarchical versus modular features in several real-life complex networks. For random scale-free topologies the extent of topological hierarchy is shown to smoothly decline with $\gamma$ -- the exponent of a degree distribution -- reaching its highest possible value for $\gamma \leq 2$ and quickly approaching zero for $\gamma>3$.