that is small for an interval of time which is exponentially large in terms of the size of the network. This low-dimensional reduction has been rigorously established in important test cases over long time scales (see [@pereira2017]). This approximation holds, roughly speaking, for two reasons. Firstly, a low degree node $i$ has a small number of connections compared to the maximum degree, and the interactions slightly perturb its statistical behaviour. Secondly, hub nodes receive a huge number of connections so that the sum of their interactions with the other nodes in the network, $\alpha \sum_j A_{ij} \bm{H}(\bm{x}_i,\bm{x}_j)$, can be approximated as the integral $\alpha k_i \int \bm{H}(\bm{x}_i,\bm{y})\rho (\bm{y}) \, d\bm{y}$ where $\rho(\bm{y})$ is the stationary distribution of typical orbits of the low degree nodes. This approximation holds up to a small fluctuation, ${\boldsymbol}\xi(t)$, which depends on the state of neighbours of the $i$th node. This fact will be key to detecting communities. The approximation described above also applies to the measured state variable $y_i(t)$. Depending on the system, we need to preprocess the data (see Methods). The processed variable is still referred to as $y_i(t)$. The Takens reconstruction technique tells us that $y_i(t+1)$ is a nonlinear function of $k+1$ past 