Fig. 3. Higher-order correlations and network asymmetries.

a, Scaling behaviour displayed by the strength–degree distribution for a single system. The adjusted exponent for this particular system gives η = 1.49. b, Distribution of exponents, P(η), for a subsample of 426 connected graphs with sizes S > 100. The average value for this distribution gives 〈η〉 = 1.50, which is consistent with the available data⁴². c, Frequency distribution of asymmetry values ϕ(i,j) between the two possible directed weights connecting a pair (i,j) of species. If we indicate these two values as ω_ij and ω_ji, the asymmetry of this pair is defined as $\varphi (i,j)=|{\omega }_{ij}-{\omega }_{ji}|/\max \left\{{\omega }_{ij},{\omega }_{ji}\right\}.$ The frequency of pairs gives an asymmetric distribution that is strongly skewed towards higher ϕ values. d, This asymmetry is also displayed in a log–log scale, markedly growing for ϕ > 0.5 values, as found in real webs⁸.