detail).

We assume the availability of a time series of observations $y_i(t)=\phi( {{\boldsymbol}x}_i(t))$, where $\phi$ is a projection to a variable on which unit interactions depend. This situation occurs frequently in applications; for example, with measurements of membrane potentials in neurons or voltages in electronic circuits. Here, we demonstrate how to obtain a model for the system by constructing an effective network from the time series $y_i(t)$.

Effective networks recover both structure and dynamics {#effective-networks-recover-both-structure-and-dynamics .unnumbered}
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To obtain an [*effective network*]{} from observations of a complex system, we combine statistical analysis, machine-learning techniques, and dynamical systems theory for networks. An effective network provides (i) local evolution laws and averaged interactions for each unit that, in combination, closely approximate the unit dynamics, and (ii) a network with the same degree distribution and communities as the original system. We use the term “effective" because, even if it does not carry information on each link and interaction in the original system, the network gathers sufficient data to reproduce the behaviour of the original network and predict its critical transitions.

[1.0]{} ![[**Reconstruction scheme with [the effective network.]{}**]{} From the time series, we build a model for the local evolution $f_i$ at each node. 