Disentangling high order effects in the transfer entropy

Transfer Entropy (TE), the main approach to determine the directed information flow within a network system, can be biased (in defect or excess), both in the pairwise and conditioned calculation, due to high order dependencies among the two dynamic processes under consideration and the remaining processes in the system which are used in conditioning. Here we propose a novel approach which, instead of conditioning the TE on all the network processes other than driver and target like in its fully conditioned version, or not conditioning at all like in the pairwise approach, searches both for the multiplet of variables leading to the maximum information flow and for those minimizing it, providing a decomposition of the TE in unique, redundant and synergistic atoms. Our approach allows to quantify the relative importance of high order effects, w.r.t. pure two-body effects, in the information transfer between two processes, and to highlight those processes which accompany the driver to build those high order effects. We report an application of the proposed approach in climatology, analyzing data from El Niño and the Southern Oscillation.