Strong correlation between circuit robustness against growth feedback and circuit topology. There are three groups of circuits, each displaying strong topological similarities within, exhibit distinct levels of robustness against growth feedback as measured by the characterizing quantity R. (a) Robustness measure R(kg = 0.6) versus Q(kg = 0) for all 425 network topologies. Circuits are color/shape-coded into three groups (green triangles, blue circles, and red diamonds) based on the rules defined in the text. The three groups of topologies display distinct levels of R(kg = 0.6) values, signifying a strong correlation between circuit robustness and topology. Only circuits with Q(kg = 0) > 300 are shown to reduce fluctuations arising from random parameter sampling. What is demonstrated is the case of an intermediate level of growth feedback with kg = 0.6 (a different value of kg has no significant effect on the results - see Fig. 7). The topologies associated with the green triangles have a high level of robustness, which can be regarded as an optimal group and is more prevalent than the optimal group identified in Fig. 5. (b) Histogram of R(kg = 0.6) the same color legends as in (a). Three distinct peaks emerge, each associated with a group of circuit topologies. (c) The shared network motif among all networks in the green group, which is highly correlated with the optimal minimal network shown in Fig. 5(b), but without the link B → B, which is necessary for the NFBL family of networks to have adaptation (Shi et al., 2017). (d) Effects of burden b for the three groups of networks, where the abscissa is the effective term of burden in the formula of growth rate Eq. (12). The circuits in the red group have larger values of 1/(1 + ⟨b⟩), suggesting that a heavier burden yields a stronger effect of the growth feedback for the red group. (e) A multilayer perceptron (MLP) for identifying the crucial connections that determine the robustness of the circuits. The circuit topology serves as the input, where 1, 0, and -1 represent activation, null, and inhibition links, respectively. The output is a predicted robustness measure, denoted as . To encourage the neural network to select as few links as possible for predicting , a l-1 regularization term, β||Win||, is incorporated into the loss function alongside the fidelity error . As a result, the feed-forward process eliminates information about the links that have little impact on circuit robustness since the corresponding Win entries automatically optimize to values close to zero. (f) Results from an ensemble of 50 MLPs, each trained with distinct initial values. Shown is the average importance of each of the nine links, which is determined by the weights in Win - see Appendix 4. The top four links with the highest importance correspond to the four links used to classify the three peaks in panel (b).