  We analyze by means of Granger causality the effect of synergy and redundancy
in the inference (from time series data) of the information flow between
subsystems of a complex network. Whilst we show that fully conditioned Granger
causality is not affected by synergy, the pairwise analysis fails to put in
evidence synergetic effects.
  In cases when the number of samples is low, thus making the fully conditioned
approach unfeasible, we show that partially conditioned Granger causality is an
effective approach if the set of conditioning variables is properly chosen. We
consider here two different strategies (based either on informational content
for the candidate driver or on selecting the variables with highest pairwise
influences) for partially conditioned Granger causality and show that depending
on the data structure either one or the other might be valid. On the other
hand, we observe that fully conditioned approaches do not work well in presence
of redundancy, thus suggesting the strategy of separating the pairwise links in
two subsets: those corresponding to indirect connections of the fully
conditioned Granger causality (which should thus be excluded) and links that
can be ascribed to redundancy effects and, together with the results from the
fully connected approach, provide a better description of the causality pattern
in presence of redundancy. We finally apply these methods to two different real
datasets. First, analyzing electrophysiological data from an epileptic brain,
we show that synergetic effects are dominant just before seizure occurrences.
Second, our analysis applied to gene expression time series from HeLa culture
shows that the underlying regulatory networks are characterized by both
redundancy and synergy.
