Fitting network measure scaling and permutation tests
This protocol is extracted from research article:
Universal scaling across biochemical networks on Earth
Sci Adv, Jan 16, 2019; DOI: 10.1126/sciadv.aau0149

For each network measure, a scaling relationship was fit as a function of the size of the LCC of the network. For each measure, three different models were tested: a power law of the form y = y0 xβ, a linear relationship of the form y = βx + y0, and a quadratic function of the form y = β1x + β2x2 + y0. For both the assortativity measures, the preferred fit was also compared with a constant y = β. The preferred model was chosen as the one that minimized cross-validation errors, according to 10-fold cross validation, across the entire dataset.

Once a model was chosen, a simulated permutation test was performed to determine whether the scaling relationship for a given attribute was the same for ecosystems and individuals or whether it was distinct (61). We took as the null hypothesis that the scaling relationship across different levels of organization is constant and used the fitted scaling parameters (for individuals and ecosystems) as the test statistic. We used fitted 1,000,000 resamples of the complete dataset to estimate the likelihood of the fit for individuals (or ecosystems) to have been drawn randomly from the complete dataset. We performed this test for both the ecosystem and individuals; if there was a difference in the estimated likelihoods, we took the greater of the two. These likelihoods are the (two-sided) P values reported in table S2. The same procedure was followed to determine the distinguishability of ecosystem networks with the randomized controls (random genome networks and random reaction networks). Random reaction networks were distinguishable from ecosystems networks for all measures, with P = 10−6.

To estimate the true scaling parameters and 95% confidence intervals, a bootstrap sample of 100,000 was used for each network attribute (61). If the permutation test allowed us to reject the hypothesis of a constant scaling relationship across individuals and ecosystems to a confidence greater than 0.01, then the scaling parameters were estimated separately for the individuals and ecosystems, otherwise the complete dataset was fit. The scaling parameters (and confidence intervals) for distinct domains were also estimated using a bootstrap of 100,000 samples. For scaling fits and confidence intervals, see data file S1.

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