![]() ![]() Mathematically speaking, a bipartite network is defined as a graph with two sets L and Γ of nodes, and a ∣L ∣ × ∣Γ ∣ matrix of connections M called bi-adjacency matrix. Bipartite networks are the natural representation for several systems, such as: social affiliation and collaboration networks, where individuals connect to the groups they are member of 10, 11 financial and commercial ownership networks, where entities are linked to the goods they own or consume 12, 13 trade networks, where economies connect to the products they export 14, 15 ecological networks, where species connect to the habitat they live in 16, 17 biological and medical networks connecting, e.g., patients and diseases 18, 19. As such, network science has gained increasing popularity in the last twenty years 6, 7, 8.Ī network is labeled as bipartite when its elements (the nodes) can be split in two disjoint sets, such that links can only exist between nodes of different sets 9. Independently of the nature of the underlying interactions, the network representation allows capturing the emergent features of these systems as well as their dynamical patterns 1, 2, 3, 4, 5. Networks are simplified yet effective models for a large class of natural, socio-economic and technological systems described by complex interaction patterns. We illustrate this procedure using data on scientific production of world countries. We thus propose a meta-validation approach that allows to identify model-specific significance thresholds for which the signal is strongest, and at the same time to obtain results independent of the way in which the null hypothesis is formulated. Instead a much better agreement is obtained for the same density of validated links. Here we systematically investigate the application of these formulations in validating the same network, showing that they lead to different results even when the same significance threshold is used. However different CM formulations exist, depending on how the constraints are imposed and for which sets of nodes. ![]() The standard approach to identify significant links is statistical validation using a suitable null network model, such as the popular configuration model (CM) that constrains node degrees and randomizes everything else. Monopartite projections of bipartite networks are useful tools for modeling indirect interactions in complex systems. ![]()
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