We investigate the relationship between complexity, information transfer andthe emergence of collective behaviors, such as synchronization and nontrivialcollective behavior, in a network of globally coupled chaotic maps as a simplemodel of a complex system. We calculate various quantities for this system: themean field, a measure of statistical complexity, the information transfer, aswell as the information shared, between the macroscopic and local levels asfunctions of the strength of a coupling parameter in the system. Our resultsshow that the emergence of nontrivial collective behavior is associated tohigher values of complexity. Little transference of information from the globalto the local level occurs when the system settles into nontrivial collectivebehavior while no information at all flows between these two scales in asynchronized collective state. As the parameter values for the onset ofnontrivial collective behavior or chaos synchronization are approached, theinformation transfer from the macroscopic level to the local level is higher,in comparison to the situation where those collective states are alreadyestablished in the system. Our results add support to the view of complexity asan emergent collective property that is absent at the local level in systems ofinteracting elements.
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