term $\xi_i(t)$ at a node $i$ depends on the state of all nodes interacting with $i$. Thus the correlation between $\xi_i$ and $\xi_j$ is proportional to the matching index (number of common neighbors) of the nodes $i$ and $j$. Thus, we can create a network that has the same statistical properties of the actual one. Further details are provided in the next section.

[**Benchmark model for the isolated dynamics.**]{} We consider heterogeneous networks of neurons described by the Rulkov model (see Methods for details). The model has two variables, $u$ and $w$, evolving at different time scales. The fast variable $u$ describes the membrane potential and $w$ the slow currents. In the following discussion, we choose the parameters such that the isolated dynamics presents a tonic spiking behaviour. In the Supplementary Materials, we also show the results for bursting dynamics, i.e. when the parameters are such that the membrane potential oscillates between two phases: a bursting phase of a rapid spiking, and a quiescent phase. We also validated our methods for other types of isolate chaotic dynamics (also in the Supplementary materials).

Revealing community structure: the rich-club motif {#revealing-community-structure-the-rich-club-motif .unnumbered}
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We apply the effective network in a wide range 