Complex networks typically display patterns that involve multiple nodes, such as motifs, hierarchies and communities. These structures can be detected as the properties of the empirical system that deviate from a benchmark model. Maximum entropy models, in particular, provide appropriate null models by constraining some properties of the network and randomising everything else. In this way, they embody a null hypothesis that these constraints can explain any other property of the system. To clarify the concept, take for example a maximum entropy null model defined by constraining property X observed in the empirical network. If the model also reproduces property Y then we know right away that Y is just a statistical consequence of X. Otherwise if Y is not reproduced, the null hypothesis is rejected and we can statistically validate Y as a salient feature of the network that is independent of X. This also implies that the empirical network is out of the statistical equilibrium of the maximum entropy ensemble defined by X.

Statistical validation is particularly useful in the context of bipartite networks, where nodes are divided into two sets such that links exist only between nodes of different sets. This is the natural representation for many system, such as a network of individuals connected with the groups they are affiliated with, or a network of financial institutions and the assets that form their portfolios. In these system, to determine the similarity of two nodes of the same set, the simplest approach is to project the network and count how many common neighbors these nodes have in the other set. However, this measure can be easily influenced by single-node variables; for instance, nodes that have high degree (many connections) in the bipartite network are likely to have more common neighbors than low-degree nodes. 

In this other paper we applied similar ideas in another context, namely to bipartite networks of countries economies and the various activities (scientific publication, patenting, and industrial production in different sectors) in which they are proficient. We put together and project different dataset to build a holistic view of the innovation system as a multilayer network of interactions among these activities. In the field of Economic Complexity, the common neighbors of two activities take the name of co-occurrences and are used as proxy of the overlap between the capabilities required to achieve a competitive advantage in both activities. In this situation the BiCM-induced null model represents the hypothesis that activities are independent and there is no capability structure behind the network, so that co-occurrences happen at random, some more likely than others just because of the ubiquity of activities and the diversification of countries. We showed that significant co-occurrences allow to identify which capabilities and prerequisites are needed to be competitive in a given activity, and even measure how much time is needed to transform, for instance, the technological know-how into economic wealth and scientific innovation, being able to make predictions with a very long time horizon.