  We consider the formalism of information decomposition of target effects from
multi-source interactions, i.e. the problem of defining redundant and
synergistic components of the information that a set of source variables
provides about a target, and apply it to the two-dimensional Ising model as a
paradigm of a critically transitioning system. Intuitively, synergy is the
information about the target variable that is uniquely obtained taking the
sources together, but not considering them alone; redundancy is the information
which is shared by the sources. To disentangle the components of the
information both at the static level and at the dynamical one, the
decomposition is applied respectively to the mutual information and to the
transfer entropy between a given spin, the target, and a pair of neighbouring
spins (taken as the drivers). We show that a key signature of an impending
phase transition (approached from the disordered size) is the fact that the
synergy peaks in the disordered phase, both in the static and in the dynamic
case: the synergy can thus be considered a precursor of the transition. The
redundancy, instead, reaches its maximum at the critical temperature. The peak
of the synergy of the transfer entropy is far more pronounced than those of the
static mutual information. We show that these results are robust w.r.t. the
details of the information decomposition approach, as we find the same results
using two different methods; moreover, w.r.t. previous literature rooted on the
notion of Global Transfer Entropy, our results demonstrate that considering as
few as three variables is sufficient to construct a precursor of the
transition, and provide a paradigm for the investigation of a variety of
systems prone to crisis, like financial markets, social media, or epileptic
seizures.
