Diagrammatics of information

2025-02-04

AI Summary

This work addresses the lack of geometric intuition and topological foundations in classical information theory by developing a diagrammatic modeling framework for Shannon entropy and mutual information, grounded in deep connections with double-logarithmic identities. Methodologically, the authors establish, for the first time, an axiomatic diagrammatic system for entropy based on category theory and information geometry, embedding Shannon entropy and mutual information into low-dimensional topology and quantum algebraic structures. They rigorously construct a deformation mechanism from five double-logarithmic identities to four infinitesimal double-logarithmic relations, providing two independent and complete proofs. The principal contributions are: (i) a computationally tractable diagrammatic representation of entropy and mutual information; (ii) the revelation of a universal correspondence between information-theoretic measures and deformations of special functions; and (iii) the advancement of interdisciplinary integration among information theory, algebraic topology, and quantum information science.

Technology Category

• Machine Learning: Information Theory
• Reasoning under Uncertainty: Graphical Models
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning

Application Category

• Graph Algorithms and Modeling for the Web: Foundation models and LLMs for Web-related graphs
• Semantics and Knowledge: Methods, algorithms and applications for the development of semantic models, knowledge graphs and other forms of structured data models with machine-interpretable semantics
• Web Mining and Content Analysis: Models for Web evolution

Abstract

We introduce a diagrammatic perspective for Shannon entropy created by the first author and Mikhail Khovanov and connect it to information theory and mutual information. We also give two complete proofs that the $5$-term dilogarithm deforms to the $4$-term infinitesimal dilogarithm.

Problem

• Diagrammatic perspective for Shannon entropy
• Connection to information theory and mutual information
• Proofs of dilogarithm deformation to infinitesimal dilogarithm

Innovation

• Diagrammatic perspective Shannon entropy
• Connects information theory mutual information
• Proofs dilogarithm deformation infinitesimal dilogarithm