More realistic models will include non-neutral processes, such as interactions that depend on species identity [32, 33], but neutral theory can act as a null model for assessing the weight of evidence for such processes.

As the dynamics are made more non-neutral, the deviations from the SNM should become stronger, and we expect the power of the test of the neutral null model to increase.

Because, for the non-neutral model in this limit, I is nothing more than a sample size—a community of size 2 I can be constructed by adding two communities of size I—this non-monotonic relationship between power and I must be due to the nonlinear role played by I in the community dynamics in the neutral null model .

For related reasons, the probability of rejecting the neutral null model was low for model IF/ LOGS when fitted to the empirical data.

For each forest, and separately for each survey year, we tested the null model that the data were generated by the SNM using the parametric bootstrap method described above.

For each data set, perform a test of the neutral null model using the parametric bootstrap method described above.

the neutral null model was rejected).

Testing the neutral null model

To calculate the p-value of our test, we compare the value of a test statistic for the test data set with values of the test statistic for data sets generated by the null model .

That modularity optimization method relies on the maximization of a benefit function Q, which measures the difference between the number of connections within each module and the expected fraction of such connections given a “null model” , that is, a statistical model of random networks.

For undirected networks, the null model traditionally corresponds to random networks constrained to have the same degree sequence as the network whose modularity is measured.

Our results are qualitatively unchanged when using layered, feed-forward networks as “null model” to compute and optimize Q (Supp.

Two different null models for calculating the modularity score.

The conventional way to calculate modularity is inherently relative: one computes the modularity of network N by searching for the modular decomposition (assigning N’s nodes to different modules) that maximizes the number of edges within the modules compared to the number of expected edges given by a statistical model of random, but similar, networks called the “null model” .

Here, we calculated the modularity Q-score with two different null models , one modeling random, directed networks and the other modeling random, layered, feed-forward networks.

The second common method is known as the Likelihood-Ratio-Test (LRT), Where one model is defined as the null model and another nested model is compared against the null model [73].

Here, the comparison is performed by where Adf denotes the difference in degrees of freedom of the null model and the nested model.

Hereby, one tests if a nested model is a valid simplification of the null model .

For a dataset generated from this null model , the p-values should be uniformly distributed from 0 to 1, exclusive: the highest Kendall’s 1‘ out of N tests should have a p-value of 1 / (N + 1), the second highest test statistic has a p-value of 2/ (N + 1), and the ith highest test statistic has a p-value of i/ (N + 1) [35].

The domain of test statistics increases very quickly with the number of data points, however, so Monte Carlo (MC) sampling, in which representations of the null model are randomly generated and evaluated, is required to estimate p-values if there are more than approximately 20 time points due to the computational cost of the generating function.

While the empirical calculation approximates the null model well, it does not fully prevent multiple hypothesis testing from weakening the ability to identify rhythmic time series.