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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)?

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Is there a way to obtain weighted random networks using degree preserving randomisation?

It isn't always possible to (non-trivially) randomise weighted edges while preserving vertices' strengths (that is, the sum of weights of edges incident with each vertex). For example, consider the weighted graph $G$ with vertex set $\{a,b,c,d,e\}$, edge set $\{\{a,b\},\{b,c\},\{c,d\},\{d,e\}\}$, and edge weights $w(\{a,b\})=1$, $w(\{b,c\})=2$, $w(\{c,d\})=3$ and $w(\{d,e\})=4$. Every rewiring of $G$ changes the strengths of at least one of $b$, $c$ and $d$. Thus, every strength-preserving "randomisation" of $G$ is isomorphic to $G$.

That said, strength-preserving randomisation may be possible in your case if some edges share the same weights. Rewiring such edges randomly will preserve vertices' strengths and (probably) deliver a non-isomorphic graph. You could do this by stratifying the edge set by edges' weights and then randomly swapping the tail vertices for pairs of edges (e.g., $v_1$ and $v_2$ for edges $\{u_1,v_1\}$ and $\{u_2,v_2\}$) in each stratum.

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)?

I'm a new user so don't have enough reputation to comment asking for clarification. If by "connections" you mean "weighted edges" then I imagine the edges you observe consistently across randomised networks will be edges that have unique weights, since these edges can't (in general) be rewired in a strength-preserving way. Such edges need not represent "significant" connections.

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