Ecological networks balance connectivity with flexibility to perturbations

Key results

1. ShownConnectance falls as a power of species richness in the pooled data and within each interaction class. The decline is shallower than the S^−1 bound derived from random-matrix stability and feasibility theory.

   Pooled fit C ∼ S^−0.65 (95% CI [−0.67, −0.64], R^2 = 0.77, p < 0.001) on the full dataset (1,569 networks in S3; Fig. 2b legend says ∼1300); food web, host–parasite, pollination and seed-dispersal fits agree within 95% CIs, as do bipartite vs non-bipartite subsets; random-matrix baseline exponent is −1. · A broad architectural tendency across ecological…; Figure legends

2. ShownA fixed number of partners per species does not explain the sparsity. Mean degree rises with richness, so total links grow superlinearly, between the constant-degree (L ∼ S) and fully connected (L ∼ S^2) limits.

   ⟨k⟩ ∼ S^0.43 (R^2 = 0.54, p < 0.001), with the same conclusion within interaction types; L ∼ S^γ with γ = 1.43 [1.41, 1.45] (R^2 = 0.93, p < 0.001); degree distributions are heavy-tailed and fitted by truncated power laws. · A broad architectural tendency across ecological…; Figure legends

3. SuggestedThe efficiency metric weighs propagation gain against loss of response diversity. By this metric, observed networks sit near a local maximum: randomly adding or deleting links mostly lowers efficiency, and a mean-field estimate of the optimal link-scaling exponent is close to the fitted one.

   Δη = ηperturbed − ηemp after adding or removing 1% of links is mostly negative (Fig. 3d, at τdiff = 1/λ2); changes of 1% to 10% in either direction also give negative Δη (100 perturbed networks per case); mean ⟨γopt⟩ = 1.42 vs fitted γ = 1.43 [1.41, 1.45]. The authors call this compatibility, not prediction. · Diffusion dynamics and the structural balance of…; Interaction perturbation experiments; Optimal scaling from thermodynamic efficiency; Figure legends

4. ShownDegree heterogeneity explains a substantial fraction of the diffusion behaviour. Random graphs matched on size and density fall short of empirical efficiency, whereas degree-preserving rewiring closely tracks it over most of the diffusion range.

   Erdős–Rényi nulls preserving size and density show systematically lower efficiency at intermediate and large τ in every interaction type; configuration-model nulls preserving degree sequence follow empirical η(τ) closely over most of the range, with small but systematic deviations in several interaction types, particularly mutualistic systems (higher empirical efficiency at intermediate τ); 100 nulls per network. · Diffusion dynamics and the structural balance of…; Thermodynamic efficiency of signal propagation; Null network models

5. ShownMutualistic networks, especially plant–pollinator and plant–ant, beat degree-preserving null efficiency more often than antagonistic networks. Networks with this excess also tend to have shorter path lengths.

   A higher fraction of mutualistic networks have ΔηCM > 0 (binomial 95% CIs, Fig. 4a), while antagonistic networks stay near CM expectations; several mutualistic classes keep positive deviations under a fixed-marginal Curveball null; ΔηCM correlates with the global-efficiency deviation ΔεCM (R^2 = 0.41, p < 0.001). The text gives no fractions. · Mutualistic networks further exploit network…; Null network models; Global efficiency; Figure legends