Covariance and correlation estimators in bipartite complex systems with a double heterogeneity

Complex bipartite systems are studied in Biology, Physics, Economics, and Social Sciences, and they can suitably be described as bipartite networks. The heterogeneity of elements in those systems makes it very difficult to perform a statistical analysis of similarity starting from empirical data. Though binary Pearson's correlation coefficient has proved effective to investigate the similarity structure of some real-world bipartite networks, here we show that both the usual sample covariance and correlation coefficient are affected by a bias, which is due to the aforementioned heterogeneity. Such a bias affects real bipartite systems, and, for example, we report its effects on empirical data from two bipartite systems. Therefore, we introduce weighted estimators of covariance and correlation in bipartite complex systems with a double layer of heterogeneity. The advantage provided by the weighted estimators is that they are unbiased and, therefore, better suited to investigate the similarity structure of bipartite systems with a double layer of heterogeneity. We apply the introduced estimators to two bipartite systems, one social and the other biological. Such an analysis shows that weighted estimators better reveal emergent properties of these systems than unweighted ones.