Towards a generalization of information theory for hierarchical partitions
This work provides a theoretical framework for studying hierarchical structures in complex systems, which is incremental as it builds on existing information theory concepts.
The authors tackled the problem of analyzing complex systems with hierarchical organization by introducing a generalization of information theory for hierarchical partitions, proving bounds and properties for hierarchical mutual information and other quantities, and validating results with numerical computations.
Complex systems often exhibit multiple levels of organization covering a wide range of physical scales, so the study of the hierarchical decomposition of their structure and function is frequently convenient. To better understand this phenomenon, we introduce a generalization of information theory that works with hierarchical partitions. We begin revisiting the recently introduced Hierarchical Mutual Information (HMI), and show that it can be written as a level by level summation of classical conditional mutual information terms. Then, we prove that the HMI is bounded from above by the corresponding hierarchical joint entropy. In this way, in analogy to the classical case, we derive hierarchical generalizations of many other classical information-theoretic quantities. In particular, we prove that, as opposed to its classical counterpart, the hierarchical generalization of the Variation of Information is not a metric distance, but it admits a transformation into one. Moreover, focusing on potential applications of the existing developments of the theory, we show how to adjust by chance the HMI. We also corroborate and analyze all the presented theoretical results with exhaustive numerical computations, and include an illustrative application example of the introduced formalism. Finally, we mention some open problems that should be eventually addressed for the proposed generalization of information theory to reach maturity.