Table 1 Scaling exponents θJ and θQ for prototypical dynamical models
From: Impact of basic network motifs on the collective response to perturbations
Model | Dynamical Equation | θJ | θQ |
|---|---|---|---|
Regulatory (\({\mathbb{R}}\)) | \(\dot{{x}_{i}}(t)=-B{x}_{i}^{a}(t)+\alpha \mathop{\sum }\nolimits_{j=1}^{N}{A}_{ji}\frac{{x}_{j}^{b}(t)}{1+{x}_{j}^{b}(t)}\) | \(\frac{1}{a}-1\) | \(-\frac{b}{a}\) |
Human (\({\mathbb{H}}\)) | \(\dot{{x}_{i}}(t)=-B{x}_{i}^{a+b}(t)+\alpha {x}_{i}^{b}(t)\mathop{\sum }\nolimits_{j=1}^{N}{A}_{ji}\left({y}_{0}-{x}_{j}^{-c}(t)\right)\) | \(\frac{1-b}{a}-1\) | \(-\frac{c}{a}\) |
Epidemics (\({\mathbb{E}}\)) | \(\dot{{x}_{i}}(t)=-B{x}_{i}(t)+\alpha \left(1-{x}_{i}(t)\right)\mathop{\sum }\nolimits_{j=1}^{N}{A}_{ji}{x}_{j}(t)\) | − 1 | − 1 |
Mutualistic (\({\mathbb{M}}\)) | \(\dot{{x}_{i}}(t)=B{x}_{i}(t)\left(1-\frac{{x}_{i}^{a}(t)}{C}\right)+\alpha {x}_{i}(t)\mathop{\sum }\nolimits_{j=1}^{N}{A}_{ji}\frac{{x}_{j}(t)}{1+{x}_{j}(t)}\) | − 1 | \(-\frac{1}{a}\) |
Population (\({\mathbb{P}}\)) | \(\dot{{x}_{i}}(t)=-B{x}_{i}^{a}(t)+\alpha \mathop{\sum }\nolimits_{j=1}^{N}{A}_{ji}{x}_{j}^{b}(t)\) | \(\frac{1}{a}-1\) | \(\frac{b}{a}\) |
Biochemical (\({\mathbb{B}}\)) | \(\dot{{x}_{i}}(t)=B-C{x}_{i}(t)-\alpha {x}_{i}(t)\mathop{\sum }\nolimits_{j=1}^{N}{A}_{ji}{x}_{j}(t)\) | − 1 | − 1 |
Inhibitory (\({\mathbb{I}}\)) | \(\dot{{x}_{i}}(t)=-B{x}_{i}(t){\left(1-\frac{{x}_{i}(t)}{C}\right)}^{2}+\alpha {x}_{i}(t)\mathop{\sum }\nolimits_{j=1}^{N}{A}_{ji}{x}_{j}(t)\) | − 1 | \(\frac{1}{2}\) |
