Decomposability of Multivariate Interactions
Naoki Miyagawa 1,2
Hiroshi Teramoto 1,3
Chun-Biu Li 1,2,4
Tamiki Komatsuzaki 1,3,4
1Research Institute for Electronic Science
Hokkaido University, Sapporo 001-0020, Japan
2Department of Mathematics
Hokkaido University, Sapporo 060-0810, Japan
3Graduate School of Life Science
Hokkaido University, Sapporo 060-0810, Japan
4Research Center for Integrative Mathematics
Hokkaido University, Sapporo 060-0812, Japan
g05041037@gmail.com, teramoto@es.hokudai.ac.jp, cbli@es.hokudai.ac.jp, tamiki@es.hokudai.ac.jp
Abstract
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.
