A Unified Mathematical Framework for Distributed Data Fabrics: Categorical Hypergraph Models

TL;DR

This paper introduces a rigorous mathematical framework for distributed data fabrics using hypergraph models and category theory, aiming to improve consistency, scalability, and relational understanding in distributed data systems.

Contribution

It develops a unified categorical hypergraph model for data fabrics, integrating algebraic, geometric, and computational methods to address key challenges in distributed data management.

Findings

Proves NP-hardness of schema matching and partitioning tasks.

Introduces spectral and symmetry-based methods for scalable data operations.

Ensures data consistency and causality under CAP and CAL theorems.

Abstract

Current distributed data fabrics lack a rigorous mathematical foundation, often relying on ad-hoc architectures that struggle with consistency, lineage, and scale. We propose a mathematical framework for data fabrics, unifying heterogeneous data management in distributed systems through a hypergraph-based structure \( \mathcal{F} = (D, M, G, T, P, A) \). Datasets, metadata, transformations, policies, and analytics are modeled over a distributed system \( \Sigma = (N, C) \), with multi-way relationships encoded in a hypergraph \( G = (V, E) \). A categorical approach, with datasets as objects and transformations as morphisms, supports operations like data integration and federated learning. The hypergraph is embedded into a modular tensor category, capturing relational symmetries via braided monoidal structures, with geometric analogies to Hurwitz spaces enriching the algebraic modeling. We prove the NP-hardness of critical tasks, such as schema matching and dynamic partitioning, and propose spectral methods and symmetry-based alignments for scalable solutions. The framework ensures consistency, completeness, and causality under CAP and CAL theorems, leveraging sparse incidence matrices and braiding actions for fault-tolerant operations.