Fitness-Based Growth of Directed Networks with Hierarchy

TL;DR

This paper introduces a simple growing network model combining degree-based preferential attachment and node fitness to explain the emergence of hierarchy and directionality in directed networks, aligning with real-world network features.

Contribution

It presents a novel model that links network growth mechanisms with hierarchy and directionality, demonstrating how these features can coexist and be predicted by fitness parameters.

Findings

Preferential attachment can induce hierarchy in networks.

Scale-free degree distributions can coexist with hierarchy.

Node fitness correlates with trophic level and predicts network incoherence.

Abstract

Growing attention has been brought to the fact that many real directed networks exhibit hierarchy and directionality as measured through techniques like Trophic Analysis and non-normality. We propose a simple growing network model where the probability of connecting to a node is defined by a preferential attachment mechanism based on degree and the difference in fitness between nodes. In particular, we show how mechanisms such as degree-based preferential attachment and node fitness interactions can lead to the emergence of the spectrum of hierarchy and directionality observed in real networks. In this work, we study various features of this model relating to network hierarchy, as measured by Trophic Analysis. This includes (I) how preferential attachment can lead to network hierarchy, (II) how scale-free degree distributions and network hierarchy can coexist, (III) the correlation between node fitness and trophic level, (IV) how the fitness parameters can predict trophic incoherence and how the trophic level difference distribution compares to the fitness difference distribution, (V) the relationship between trophic level and degree imbalance and the unique role of nodes at the ends of the fitness hierarchy and (VI) how fitness interactions and degree-based preferential attachment can interplay to generate networks of varying coherence and degree distribution. We also provide an example of the intuition this work enables in the analysis of a real historical network. This work provides insight into simple mechanisms which can give rise to hierarchy in directed networks and quantifies the usefulness and limitations of using Trophic Analysis as an analysis tool for real networks.