  The ever increasing availability of data demands for techniques to extract
relevant information from complex interacting systems, which can often be
represented as weighted networks. In recent years, a number of approaches have
been proposed to extract network backbones by assessing the statistical
significance of links against null hypotheses of random interaction. Yet, it is
well known that the growth of most real-world networks is highly non-random, as
past interactions between nodes typically increase the likelihood of further
interaction. Here, we propose a network filtering methodology based on a family
of null hypotheses that can be calibrated to assess which links are
statistically significant with respect to a given network's own heterogeneity.
We design such family of null hypotheses by adapting the P\'olya urn, a simple
one-parameter combinatorial model driven by a self-reinforcement mechanism, to
a network setting. We provide a full analytical description of our filter, and
show that it retains or discards links based on a non-trivial interplay between
their own local importance and the importance of the nodes they belong to. We
prove that the widely used disparity filter can be recovered as a
large-strength approximation for a specific value of the P\'olya filter's
parameter, and illustrate our findings by applying the filter to two large
network datasets.
