rules are similar and display a chaotic evolution, thus, confirming that the similarity analysis by the functional networks would provide little insight into the reconstruction. Having the local rules $g_i$, we can decompose the time series in terms of a low-dimensional deterministic part and a fluctuation term $\xi_i$, which depends on the neighbours of the $i$th node. One of our main key points is that these fluctuations allow us to reveal the community structure.

Indeed, if nodes $i$ and $j$ interact with the same nodes, they are subject to the same inputs and the covariance Cov$(\xi_j,\xi_i)$ is high. If not, Cov$(\xi_i,\xi_j)$ is nearly zero due to the decay of correlations in the deterministic part. Therefore, Cov$(\xi_j,\xi_i)$ is high when nodes $i$ and $j$ have high [*matching index*]{} (high fraction of common connections). It is crucial that the correlation analysis is restricted to the dynamical fluctuations $\xi_i$. Indeed, since the variance of the deterministic part of $y_{i}$ is larger than that of the small fluctuations $\xi_i$, performing a direct correlation analysis between $y_{i}$ and $y_{j}$ hides all the contributions to the correlations coming from the covariance between $\xi_i$ and $\xi_j$. Consequently, the correlation of the deterministic part is close to zero 