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

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.