## The method in one line

Take a network and filter out all links with the exception of over-expressed (or under-expressed) ones with respect to a randomized null model

## The basic idea

This is a theoretical and algorithmic methodology designed to filter a complex network to its backbone structure. It can be performed both on unipartite and bipartite networks. In the bipartite case the method provides a filter of the projected network, either on the first or on the second module. The filtering is done by statistically comparing the input network with a randomized version of the same network (the Null model), obtained by fixing some properties of the real network (strength/degree distribution) and by letting links to be drawn completely at random once conditioned on the imposed constraints. The statistical filter removes all the links that come with high probability in the randomized version of the network. Namely, if one link is very likely to be there (or to have the same or greater weight) in the null model then the existence of that link (or the size of that weight) is just a statistical consequence of the general structure of the network (i.e.Â the degree distribution) and not a feature peculiar to that specific network.

## An example on Tuscan Mobility

Below you see an example of unipartite weighted network. The graph represents the traffic fluxes in the Tuscany region in Italy. The color of nodes identifies different Administrative areas. Edges have the color of their target node. Data are collected from GPS vehicle tracking over a time interval of roughly 1 month and a half. The network as 29569 links and 287 nodes. The node size is proportional to the PageRank score of the node. As you may see, the network is pretty dense, with a lot of long-distance links as well. However, colors rouhgly clusters in connected regions, meaning that intra-area traffic is dominant over inter-area traffic, as expected.

Different statistical methods can be applied for validation, some more drastic, typically removing a lot of links, controlling the Family-wise Error Rate (Bonferroni correction), and some more conservative, controlling the False Discovery Rate (Benjamini-Hochberg-Yekuteli correction).