I've been using the backbone network extraction method outlined in this paper: http://www.pnas.org/content/106/16/6483.abstract
Basically, the authors propose a method based in statistics that produces a probability, for each edge in the graph, that the edge could have happened just by chance. I use the typical statistical significance cutoff of 0.05.
I've been applying this method to several real-world networks, and interestingly some networks end up with no edges as significant. I'm trying to understand what this entails for the network. The only other time I've applied the method to a network and had no edges come out as significant was when I applied the method to random networks that I generated, which is exactly what we'd expect.
As an example real world network, you may have seen the recent network visualization that went on The Economist showing the polarization of the U.S. Senate in the past 25 years: http://www.economist.com/news/united-states/21591190-united-states-amoeba. I applied the backbone network extraction method to those networks and no edges came up as significant. Even though the raw edges apparently show preferential attachment and clustering, is this just by chance? Is the Senate voting network network essentially random?