The mathematics of non-linear metrics for nested networks

TL;DR

This paper analytically investigates the mathematical properties of the non-linear fitness-complexity metric and a new variant, revealing their convergence behaviors and potential for improved node importance rankings in nested bipartite networks.

Contribution

It provides the first rigorous analysis of the fitness-complexity metric and introduces a new variant, the minimal extremal metric, with exact solutions for perfectly nested networks.

Findings

Exact expressions for node scores in perfectly nested networks.

The minimal extremal metric can outperform the fitness-complexity metric with reliable data.

Convergence properties of the metrics are explained through derived formulas.

Abstract

Numerical analysis of data from international trade and ecological networks has shown that the non-linear fitness-complexity metric is the best candidate to rank nodes by importance in bipartite networks that exhibit a nested structure. Despite its relevance for real networks, the mathematical properties of the metric and its variants remain largely unexplored. Here, we perform an analytic and numeric study of the fitness-complexity metric and a new variant, called minimal extremal metric. We rigorously derive exact expressions for node scores for perfectly nested networks and show that these expressions explain the non-trivial convergence properties of the metrics. A comparison between the fitness-complexity metric and the minimal extremal metric on real data reveals that the latter can produce improved rankings if the input data are reliable.