Under the assumption that such rules change from node to node depending on their connectivity, we estimate the coupling function. Using the fluctuations of the time series with respect to the low-dimensional rules, we recover the community structures. Gathering all this information, we obtain an effective network which is a model for the original system that can be used to predict critical transitions](fig3_alt4_Scheme_reduced.pdf "fig:"){width="\linewidth"}

. \[RC\]

Using our assumptions for the network and local dynamics, we can show that the evolution at each node will have low-dimensional excursions over finite time scales. In particular, the approximated evolution rule at node $i$ has a low dimensional description over large time scales given by . More precisely, the evolution rule at node $i$ is given by $$\begin{aligned}
{\boldsymbol}G_i({\boldsymbol}x_i)  = {\boldsymbol}F_i({\boldsymbol}x_i) + \beta_i {\boldsymbol}V({\boldsymbol}x_i(t)) \nonumber\end{aligned}$$ where ${\boldsymbol}F_i \approx {\boldsymbol}F$ is the isolated dynamics, $\beta_i = \alpha k_i$ is the rescaled degree, and ${\boldsymbol}V$ is the averaged mean-field effect of the interaction function (effective coupling function) that takes into account the cumulative effect of interactions on node $i$. The true dynamics ${\boldsymbol}x_i(t)$ slightly deviates from this rule and is given by $$\begin{aligned}
{\boldsymbol}x_i(t+1) = {\boldsymbol}G_i({\boldsymbol}x_i(t)) + {\boldsymbol}\xi_i(t) \nonumber\end{aligned}$$ where ${\boldsymbol}\xi_i(t)$ is a fluctuation term 