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. 2016 Aug 24;7:12573. doi: 10.1038/ncomms12573

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Copyright © 2016, The Author(s)

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Frequency distributions of eigenvalues of real and permuted food webs : (a) permutation tests H1 to H4, (b) permutation tests H5 to H8. Eigenvalues are on a logarithmic scale and dimensionless. Permutation tests were carried out 1,000 times for each food web. Eigenvalue distributions were smoothed using a kernel density estimate of 0.28. (c) Stability of real and permuted food webs related to complexity (permutation tests H1–H4). Stability is measured as Re(λ_max) and corresponds to complexity. Statistics of the linear regression between complexity and stability: real food webs (slope=0.005, P=0.02, R²=0.04), H1: random row structure (slope=0.003, P=0.38, R²=0.002), H2: random topology (slope=0.006, P=0.2, R²=0.006), H3: random pairwise correlation (slope=0.06, P=10⁻⁷, R²=0.2), H4: random interaction strength distribution (slope=0.24, P<10⁻¹⁶, R²=0.65). (d) Stability of permuted food webs related to complexity (permutation tests H5–H8). Statistics of the linear regression between complexity and stability: H5: empirical distribution of interaction strengths only (slope=0.06, P=10⁻⁹, R²=0.25). H6: pairwise correlation only (slope=0.3, P<10⁻¹⁶, R²=0.63). H7: topology only (slope=0.32, P<10⁻¹⁶, R²=0.6). H8: random food webs (slope=0.33, P<10⁻¹⁶, R²=0.66).

Inline graphic — Frequency distributions of eigenvalues of real and permuted food webs : (a) permutation tests H1 to H4, (b) permutation tests H5 to H8. Eigenvalues are on a logarithmic scale and dimensionless. Permutation tests were carried out 1,000 times for each food web. Eigenvalue distributions were smoothed using a kernel density estimate of 0.28. (c) Stability of real and permuted food webs related to complexity (permutation tests H1–H4). Stability is measured as Re(λ_max) and corresponds to complexity. Statistics of the linear regression between complexity and stability: real food webs (slope=0.005, P=0.02, R²=0.04), H1: random row structure (slope=0.003, P=0.38, R²=0.002), H2: random topology (slope=0.006, P=0.2, R²=0.006), H3: random pairwise correlation (slope=0.06, P=10⁻⁷, R²=0.2), H4: random interaction strength distribution (slope=0.24, P<10⁻¹⁶, R²=0.65). (d) Stability of permuted food webs related to complexity (permutation tests H5–H8). Statistics of the linear regression between complexity and stability: H5: empirical distribution of interaction strengths only (slope=0.06, P=10⁻⁹, R²=0.25). H6: pairwise correlation only (slope=0.3, P<10⁻¹⁶, R²=0.63). H7: topology only (slope=0.32, P<10⁻¹⁶, R²=0.6). H8: random food webs (slope=0.33, P<10⁻¹⁶, R²=0.66).