Fig. 1: Example cases of simplified analytical inferences.
From: Deep learning resilience inference for complex networked systems
We perform case studies on networks with mutualistic (parameter setting: B = 0.1, C = 1, K = 5, D = 5, E = 0.9, H = 0.1) (a), gene regulatory (parameter setting: f = 1, h = 2, B = 1) (b), and neuronal (parameter setting: μ = 3.5, δ = 2) (c) dynamics. Specifically, we first plot the 1-D resilience function condensed by GBB from the original high dimensional equations, where we show the stable equilibrium of xeff (Supplementary Equation (11)) given βeff (Supplementary Equation (12)). We analyze pairs of networks that have similar βeff under GBB framework but different assortativity measured by degree correlation coefficient r. Network (I),(II),(V), and (VI) are all inferred as non-resilient by the classic GBB framework because \({\beta }_{{{\rm{eff}}}} \, < \, {\beta }_{{{\rm{eff}}}}^{c}\)(d, f). On the contrary, Network (III) and (IV) are inferred as resilient with \({\beta }_{{{\rm{eff}}}} \, > \, {\beta }_{{{\rm{eff}}}}^{c}\)(e). We further simulate the node activities using the corresponding dynamics for 100 times with varying initial conditions (Dynamics and datasets in Methods) on each network, and employ kernel density estimation (KDE) plot to visualize the distribution of its stable states. However, the ground truth simulations show Network (II), (III) and (VI) are resilient because they have a unique, non-trivial stable states (〈x〉 > 0), while Network (I), (IV) and (V) are non-resilient for having more than 1 stable states or only 1 trivial stable state (〈x〉 = 0) (g–i). Therefore, Network (II), (IV), (VI) are inaccurately inferred by the classic analytic model GBB, and they all have negative or positive assortativity.
