I need to analyze a correlation network where I do not know if scale-free properties are expected. Actually, I would like to know if the network has a power-law decay of the degree distribution (scale-free). Usually, a soft threshold is selected which gives such a power-law distribution, under the prior assumption that the network is scale-free. What if we cannot make that prior assumption? Are there alternative thresholding methods, so we can test for scale-free topology (or other tail distributions, like exponential) without prior bias?

The network I am working with is made of correlations between the abundances of bacteria in an ecosystem, as measured with metagenomics. The correlations are calculated from the relative abundance of each different bacteria of several soil samples. We use partial correlations to correct for confounding factors. We do not know if scale-free topology is expected.

I don't have a well-founded answer to your scale-free topology question; in my opinion people overemphasize scale free topology in the context of correlation networks built from noisy data.

As a practical matter though, if you're constructing a correlation network and are not sure whether the scale-free topology criterion should be used at all, you can follow the table of suggested soft-thresholding powers given in WGCNA FAQ. The basic idea behind soft-tresholding is to suppress correlations that are likely noisy. If you assume Gaussian noise, the typical noise correlation is roughly proportional to 1/sqrt(m), m being the number of samples. Roughly speaking, you want a power of beta such that when you raise the typical noise correlation to power beta and multiply it by the number of variables (genes), you get less than the a strong correlation, close to 1, raised to the same power. This thinking, and increasing the beta a bit to account for the fact that in reality, the noise is not Gaussian, will basically give you the table in WGCNA FAQ.