Structure of transfer entropy

In current methods for causality detection, a causal relationship is deduced from its effect. Even the causality keeps unchanged, its effect may display complicated behavior, i.e., it is sometimes very strong, sometimes very weak or even absent, depending strongly on the present and historical states of and the coupling strength between the causal and influential variables. This complicated behavior of the causal effect is called the structure of causality, which is merged completely into a scalar quantity in the statistical procedure shared by the causality detection methods, i.e., it is lost. In this work, this structure is preserved and described by the transitions from all the historical states to all the present states, which form a state transition network. Technically, the structure of transfer entropy is defined as a typical example for this approach, whose performance is shown by means of the coupled logistic maps and the coupled Hénon maps. It is found that a small fraction of the historical states and a small number of the transitions from historical to present states carry the substantial information, called accordingly the dominant states and the dominant transitions, all of which form a dominant state transition network. Structures of causalities in several empirical cases are then shown in detail.