Ranking nodes in bipartite systems with a non-linear iterative map

TL;DR

This paper introduces a flexible non-linear iterative map for ranking nodes in bipartite networks, unifying various importance measures, optimizing rankings for ecological and structural analysis, and revealing a phase transition phenomenon.

Contribution

It presents a novel non-linear iterative method that generalizes multiple importance measures and demonstrates superior performance in ecological network analysis and structural property evaluation.

Findings

Outperforms existing ranking algorithms in ecological network tasks

Reveals a phase transition at a critical parameter value

Exhibits a unique 'triangular' pattern near the phase transition

Abstract

Ranking nodes in networks according to a defined measure of importance is an extensively studied task, with applications in ecology, economic trade networks, and social networks. This paper introduces a method based on a non-linear iterative map to evaluate node relevance in bipartite networks. By tuning a single parameter $\gamma$, the method captures different concepts of node importance, including established measures like degree centrality, eigenvector centrality and the fitness-complexity ranking. The algorithm's flexibility allows for efficient ranking optimization tailored to specific tasks, outperforming state-of-the-art algorithms. We apply this method to ecological mutualistic networks, where ranking quality can be assessed by the extinction area - the rate at which the system collapses when species are removed in a certain order. The map with the optimal $\gamma$ value surpasses existing ranking methods on this task. Additionally, our method excels in evaluating nestedness, another crucial structural property of ecological systems, requiring specific node rankings. Finally, we explore theoretical aspects of the map, revealing a phase transition at a critical $\gamma$ dependent on the data structure that can be characterized analytically for random networks. Near the critical point, the map exhibits unique features and a distinctive "triangular" packing pattern of the incidence matrix.