Decomposability Of Multivariate Interactions

Systems in nature, composed of many microscopic components, exhibit several distinctive global patterns. Can we understand the emergent patterns in terms of the components? One possible device to address such a question is to scrutinize the "interactions" among these components, from which the global behavior arises. In this paper, we introduce and generalize the information- theoretic quantity called connected information. It provides us with a measure of many- body interactions buried in complex systems. While the original connected information is formulated globally to include all contributions from the microscopic components, we formulate decomposition rules for the connected information to capture local interactions. The implication of our results will also be discussed in relation to the identification of local functional modules in neural systems based on experimental observations.