  In many data sets, crucial information on the structure and temporality of a
system coexists with noise and non-essential elements. In networked systems,
for instance, some edges might be non-essential or exist only by chance.
Filtering them out and extracting a set of relevant connections, the "network
backbone", is a non-trivial task, and methods put forward until now do not
address time-resolved networks, whose availability has strongly increased in
recent years. We develop here such a method, by defining an adequate temporal
network null model, which calculates the random chance of nodes to be connected
at any time after controlling for their activity. This allows us to identify,
at any level of statistical significance, pairs of nodes that have more
interactions than expected given their activities: These form a backbone of
significant ties. We apply our method to empirical temporal networks of
socio-economic interest and find that (i) at given level of statistical
significance, our method identifies more significant ties than methods
considering temporally aggregated networks, and (ii) when a community structure
is present, most significant ties are intra-community edges, suggesting that
the weights of inter-community edges can be explained by the null model of
random interactions. Most importantly, our filtering method can assign a
significance to more complex structures such as triads of simultaneous
interactions, while methods based on static representations are by construction
unable to do so. Strikingly, we uncover that significant triads are not
equivalent to triangles composed by three significant edges. Our results hint
at new ways to represent temporal networks for use in data-driven models and in
anonymity-preserving ways.
