Information theory is often utilized to capture both linear as well as nonlinear relationships between any two parts of a dynamical complex system. Recently, an extension to classical information theory called partial information decomposition has been developed, which allows one to partition the information that two subsystems have about a third one into unique, redundant and synergistic contributions. Here, we apply a recent estimator of partial information decomposition to characterize the dynamics of two different complex systems. First, we analyze the distribution of information in triplets of spins in the 2D Ising model as a function of temperature. We find that while redundant information obtains a maximum at the critical point, synergistic information peaks in the disorder phase. Secondly, we characterize 1D elementary cellular automata rules based on the information distribution between neighboring cells. We describe several clusters of rules with similar partial information decomposition. These examples illustrate how the partial information decomposition provides a characterization of the emergent dynamics of complex systems in terms of the information distributed across their interacting units.
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