I have a network in which nodes are highly interconnected (250 nodes where 90% of the nodes have degree = 249). The connections are weighted with a normalised index that goes from 0 to 1, where 1 means a strong connection and values close to 0 identify weak connections. The weight distribution of the network is right-skewed, with most edges having proximity close to 0. I am trying to set a non-arbitrary threshold to decide which edges to retain; in the extant literature it seems a consolidated practice to choose a threshold arbitrarily and I can't find any standard method that allows to set a threshold of this kind (for instance, basing on the weight distribution of the edges.
Is there an empirical method to set such threshold?
I am also using degree-preserving randomisation tests in order to compare the network-level characteristics to the distribution of the same characteristics extracted from the randomised graphs. However, the randomised networks are unweighted.
Is there a way to obtain weighted random networks using degree preserving randomisation? And secondly, is there a statistical test I can perform in order to select the "significant" connections (those that are observed consistently both in the empirical network and randomised networks)?