Scale invariance and statistical significance in complex weighted networks

Most networks encountered in nature, society, and technology have weighted edges, representing the strength of the interaction or association between their vertices. Randomizing the structure of a network is a classic procedure used to estimate the statistical significance of properties of the network such as transitivity, centrality, and community structure. Randomization of weighted networks has traditionally been done via the weighted configuration model (WCM), a simple extension of the configuration model, where weights are interpreted as bundles of edges. It has previously been shown that the ensemble of randomizations provided by the WCM is affected by the specific scale used to compute the weights, but the consequences for statistical significance were unclear. Here we find that statistical significance based on the WCM is scale dependent, whereas in most cases results should be independent of the choice of the scale. A two-step approach, originally introduced for network reconstruction, in which one first randomizes the structure and then the weights, with a suitable distribution, restores scale invariance and allows us to conduct unbiased assessments of significance on weighted networks.