  Reconstruction of the network interaction structure from multivariate time
series is an important problem in multiple fields of science. This problem is
ill-posed for large networks leading to the reconstruction of false
interactions. We put forward the Ergodic Basis Pursuit (EBP) method that uses
the network dynamics' statistical properties to ensure the exact reconstruction
of sparse networks when a minimum length of time series is attained. We show
that this minimum time series length scales quadratically with the node degree
being probed and logarithmic with the network size. Our approach is robust
against noise and allows us to treat the noise level as a parameter. We show
the reconstruction power of the EBP in experimental multivariate time series
from optoelectronic networks.
