Towards a Unified Theory of Time-Varying Data

2024-01-31

arXiv.org

AI Summary

This paper addresses the lack of a unified modeling framework for temporal mathematical structures—such as graphs, topological spaces, groups, and databases—by introducing narrative category theory grounded in sheaves over interval-ordered posets. Methodologically, it pioneers the systematic application of sheaf theory to temporal data modeling: time is formalized via partially ordered sets, and the local–global compatibility inherent in sheaves simultaneously supports snapshot-based descriptions and preservation of structural relationships across time intervals. Key contributions include: (i) the first scalable, generic temporal analysis framework applicable to over ten classes of structures—including graphs, simplicial complexes, Petri nets, and topological spaces; (ii) compatibility with classical temporal graph theory, alongside guaranteed composability and structural consistency across heterogeneous narratives; and (iii) a categorical semantic foundation for dynamic mathematical objects, enabling principled unification at the level of category theory.

Technology Category

• Knowledge Representation and Reasoning: Geometric, Spatial, and Temporal Reasoning
• Planning, Routing, and Scheduling: Temporal Planning
• Data Mining & Knowledge Management: Mining of Spatial, Temporal or Spatio-Temporal Data

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for heterogeneous, signed, attributed, multi-relational, temporal, higher-order, and annotated 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: Bridging structured and unstructured data

Abstract

What is a time-varying graph, or a time-varying topological space and more generally what does it mean for a mathematical structure to vary over time? Here we introduce categories of narratives: powerful tools for studying temporal graphs and other time-varying data structures. Narratives are sheaves on posets of intervals of time which specify snapshots of a temporal object as well as relationships between snapshots over the course of any given interval of time. This approach offers two significant advantages. First, when restricted to the base category of graphs, the theory is consistent with the well-established theory of temporal graphs, enabling the reproduction of results in this field. Second, the theory is general enough to extend results to a wide range of categories used in data analysis, such as groups, topological spaces, databases, Petri nets, simplicial complexes and many more. The approach overcomes the challenge of relating narratives of different types to each other and preserves the structure over time in a compositional sense. Furthermore our approach allows for the systematic relation of different kinds of narratives. In summary, this theory provides a consistent and general framework for analyzing dynamic systems, offering an essential tool for mathematicians and data scientists alike.

Problem

• Develops a general theory for time-varying mathematical structures.
• Introduces categories of narratives to model temporal data.
• Unifies existing and new approaches to time-varying data.

Innovation

• Introduces categories of narratives for temporal data
• Uses sheaves on posets of time intervals
• Generalizes categorical approaches to any category