Figures and data
Inference of the dgLV generative model is performed by a moment matching optimization procedure. We aim to infer the free parameters θ = (μ, σ, T, λ) so that to match the order parameters and the average carrying capacity (h, q0, qd, K). This procedure allows us to extract ecological dynamics information for cross-sectional data of healthy (blue) and diseased (red) microbiomes, which are conceptualized as independent disorder realizations.
Order parameters inferred from the data using Eqs. (2.2). Panel A) h B) q0 and C) qd in healthy (blue) and unhealthy (red) cohorts. The gray points show the values for the corresponding null models where sample labels have been randomized. Standard deviations have been computed over 5000 realizations.
Panel a: inferred T (demographic noise strength) and σ (interactions heterogeneity) for healthy (blue) and diseased (red) microbiomes are clustered. Darker dots correspond to better solutions (i.e., solutions with a lower value of the cost function C), while the two points with hexagonal markers correspond to the best two (healthy and diseased, respectively) solutions. In the first panel inset, we also show (in log-log scale) the SADs corresponding to each solution. To have a more concise representation, we present each SAD fixing the disorder to its average 
The replicon eigenvalue corresponding to each solution of our optimization procedure (shaded dots). The solid hexagon represents the replicon corresponding to the best solutions that minimize the error in predicting the order parameters of the theory (minimum C). The two investigated microbiome phenotypes (healthy in blue, diseased in red) are significantly different. In particular, diseased microbiomes are closer to the marginal stability of replica-symmetric ansatz (grey horizontal line). Panel b: Solutions of the moment-matching objective function are shown as a function of ψ and m, which in turn depend on the SAD parameters (see main text). Healthy (blue) and diseased (red) microbiomes appear to be clustered. Therefore, distinct ecological organization scenarios (strong neutrality/emergent neutrality) emerge. Darker dots correspond to solutions with lower values of the cost function, while hexagonal markers correspond to the two best solutions.
