How null-model constraints affect statistical validation in projected bipartite networks
Statistical validation of projected bipartite networks depends critically on the null model adopted to describe random co-occurrences. Although several null models have been proposed, their comparison has mainly focused on the validated backbones they produce rather than on the statistical assumptions underlying their construction. Here we compare four widely used null models - the microcanonical configuration model generated by the Curveball algorithm, the Bipartite Configuration Model (BiCM), the Bipartite Partial Configuration Model (BiPCM), and the Hypergeometric approximation - using three empirical bipartite systems from comparative genomics, international trade and food science. We use the statistically validated links obtained from the microcanonical bipartite configuration model as the reference benchmark for assessing performances of the other three models. Our central result is that the statistical consequences of relaxing null-model constraints cannot be understood solely from the constraints themselves but must be analyzed through the probability distribution induced for the co-occurrence statistic. In particular, the combined behaviour of the expectation and variance largely explains the observed differences among the statistically validated backbones. We further derive a leading-order sparse approximation for the BiCM expectation, showing that the first correction to the Hypergeometric prediction is controlled by the degree heterogeneity of the non-projected layer. Surprisingly, despite neglecting this heterogeneity, the Hypergeometric model accurately reproduces the co-occurrence variance of the microcanonical ensemble across all datasets. Our results suggest that null models should be compared not only according to the constraints they preserve but also according to the statistical consequences that these constraints induce on the distribution of the test statistic.